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Mazda3 SkyActiv-X Review
By Andrew Maclean
Just when we all thought the petrol engine was heading for extinction
Mazda has developed an innovative solution that could nurture it enough to provide a new lease on life.
While it ultimately might not be enough to prevent it from the meteorite of electric cars that will likely rule the planet in decades to come, the Japanese brand has mutated the petrol engine in a way that makes it cleaner, leaner and more efficient than ever before without compromising performance or everyday driveability.
Under what Mazda calls SkyActiv-X technology, and now available in the latest-generation Mazda3 hatch and sedan, the petrol engine has what is known as Spark Controlled Compression Ignition which combines the best attributes of an economical diesel-like combustion cycle with the power and performance of a lean-burning conventional petrol engine, with the net result being a significant improvement in fuel consumption.
A diesel engine operates by igniting the fuel under extreme compression, which generally means it achieves maximum energy from each explosion and can therefore use less fuel for a bigger bang. The downside is that most diesel engines are noisier, less refined and cannot rev as high as petrol engines. And without a turbo charger – which is rare these days anyway – they don't produce a lot of power.
A conventional petrol engine, on the other hand, uses a spark plug, generally located at the top of the combustion chamber, to ignite the fuel. And because of its location, even with advanced fuel injection systems, the explosion isn't as consistent and sometimes not every joule of fuel is actually burnt which is wasting energy and increasing harmful emissions.
So what Mazda has done is take the best bits from each and stitch them together – while also linking the engine with a mild hybrid system and a 24-volt lithium-ion battery for even greater efficiency – which makes the technology more affordable, lighter and (slightly) less complex than a battery-powered hybrid. It also takes up no more room under the bonnet than a normal engine, so there are no compromises to storage space.
Essentially, the SkyActiv-X engine – a fairly normal 2.0-litre naturally-aspirated four-cylinder – operates at low engine speeds by injecting a fine-spray of fuel into each cylinder on the intake stroke followed by another small ball of fuel, pointed directly at the spark plug on the compression stroke. Like a golf ball squashing into a sponge, when the spark plug ignites the richer mixture at the top the finer, lower layer of fuel is so heavily compressed that it rapidly combusts like a diesel engine and creates a more uniform explosion. What that means is it requires less fuel to produce the equivalent amount of power and subsequently improves overall economy.
The trickery extends to the fact that under heavy acceleration and at high engine speeds, where the pressure loads of the compression ignition cycle would be too immense for long-term reliability, the engine seamlessly switches to a conventional spark ignition cycle.
At the moment, the SkyActiv-X engine is exclusively available in the range-topping Mazda3 X20 Astina, which cost $40,590 (plus on-roads), but the technology will be spread further across the Mazda range, starting with the CX-30 small SUV that arrives in Australian showrooms soon.
For the same money, you can choose between either the bold five-door hatchback (as tested here) or the sleek four-door sedan, both of which come with a six-speed manual as standard or an optional six-speed automatic and drive the front wheels.
While the body styles are different, they share an identical level of standard equipment that includes leather interior trim with heated front seats, dual-zone air conditioning, a part-digital instrument cluster and a widescreen dash-mounted display with sat nav, Bluetooth and smartphone mirroring for Apple and Android mobile devices.
There's a full suite of active safety features too, such as automated emergency braking, active cruise control, blind spot detection, lane keeping assistance, front and rear cross-traffic warnings and a 360-degree reverse camera.
While it's difficult to explain precisely how the SkyActiv-X technology works, it doesn't require any modification to your behaviour in order to reap its benefits as it drives just like any other normal car.
With the engine producing 132kW of power and 224Nm of torque, it feels zippy around town and engaging and fun to drive through the twisty stuff and yet is even more leisurely out on the open road than any conventional alternative.
It doesn't sound any different to a normal petrol engine either and it is just as refined with nothing noticeable in the way it switches between the two different combustion techniques. If anything, it feels stronger than a similar-sized petrol engine at low engine speeds with more immediate urgency, making it more effortless to drive in heavy traffic and over long distances.
The integrated starter motor hybrid system, which switches off the engine when stopped at traffic lights and adds a small battery-powered boost when getting going again, operates seamlessly and further enhances the fuel-saving measures.
Speaking of which, Mazda claims the SkyActiv-X Astina consumes an average of 5.3L/100km with the manual transmission (and 5.5L/100km with the automatic), which it says is around 10 percent more efficient than a conventional petrol engine of the same capacity. It's not that hard to match that figure if you regularly mix steady-state highway cruising into your everyday driving behaviour, but expect to see returns between 6-7L/100km – which is still a reasonable result – if you mainly drive in built-up areas.
As for the rest of the package, the latest-generation Mazda3 is a classy little machine. The interior quality is a step beyond the mainstream and verges on offering a genuine European-level of luxury and refinement. The dashboard's minimalistic design is both stylish and easy to use with simple controls for the ventilation and clear instruments in front of the driver, while Mazda's latest infotainment interface is significantly better to navigate through its functionality.
The front seats are super comfortable with plenty of adjustment to suit a wide variety of drivers, but the hatchback's swooping exterior design does compromise the comfort of those in the back with the high window line making the rear space feel tighter than it is while also introducing massive blind spots in rearward vision for the driver.
The Mazda3 was already a benchmark within the small hatchback segment – remaining one of Australia's most popular passenger cars – and the introduction of the SkyActiv-X engine improves on its position, even though the fuel-saving technology is restricted to those willing to pay a premium for the flagship model.
There's no doubt it works, and the benefits are real, but is it a big enough leap to save the petrol engine from extinction? Probably not – at least on its own – but when Mazda moves forward and links SkyActiv-X to proper plug-in powertrains it will create the ultimate hybrid, blending the best of battery, petrol and diesel power together.
2020 Mazda3 X20 Astina Hatch
Price: From $40,950 (plus on-roads)
Engine: 2.0-litre four-cylinder
Power: 132kW/224Nm
Transmission: Six-speed manual (or automatic) transmission
Fuel consumption: 5.3L/100km (man) / 5.5L/100km (auto)
andrew maclean, mazda, review
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Learn digital painting : Rhoda Draws!
Learn digital painting technique from critically acclaimed books featured on the Corel website and written by….ME!
Using Corel Painter software and a WACOM tablet I have been creating unique digital artwork one brush-stroke at a time for 23 years.
With Corel Painter software, a pressure sensitive tablet from Wacom Technologies, and a reasonably powerful computer, you can imitate virtually any "natural" media. Then just buy one of my books and… practice, practice, practice.
Start by scribbling, so you can get familiar with the Wacom tablet and learn to manage the brushes, colors, textures and other art, materials available in Painter.
Then do a simple drawing of a pear using Painter's tracing paper feature. You'll move on to basic landscapes and city scenes. Advanced projects include illustration, life drawing, portraiture and abstract painting.
By the time you finish my Painter book you will have learned the principles of pixel-based digital art and mastery of Painter's essential features. You will also have developed many fundamental skills needed for drawing and painting in any medium.
See sample pages from my Digital Painting Fundamentals with Corel Painter books on Amazon.com.
Want to see a live digital caricature artist in action? Hire Rhoda Draws a Crowd! to entertain at your event.


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Biography of William I. Main
A native and lifelong resident of Washington County, Mr. Main has been for many years the owner of one of the leading jewelry stores and watch repair establishments in Wakefield. He is also successfully engaged in the radio business and in both of these lines he enjoys a very large trade, his success being the result of his energy, his well known integrity, and his strict adherence to the highest business principles. He enjoys a very large acquaintance in the community, where he is generally regarded as one of the most popular, substantial, and useful citizens.
William I. Main was born at Arcadia, Washington County, Rhode Island, July 20, 1887, a son of William G. and Emma Frances (Whitman) Main. His father, who was born at Norwich, Connecticut, was for many years and until his death success fully engaged in the ice cream manufacturing business in Wakefield, being the proprietor of Main's Ice Cream Company. Mr. Main's mother, who is still living and a resident of Wakefield, was born in Rhode Island. Having been educated in the public grammar and high schools of South Kingstown, Washington County, Mr. Main, at the age of seventeen years, entered the employ of George H. Taylor & Company, well-known jewelers of Providence, with which he remained for four years. During this period he learned the jeweler's and watch-maker's trade in the most thorough manner and, in 1907, he established himself in the jewelry and watch-repairing business at Wakefield under his own name. He has continued in it ever since then and his establishment for many years has been one of the most popular and best known in the city. Since 1922 he has also been engaged in selling radios and in servicing these instruments, meeting with marked and well deserved success in this undertaking as well. He is a member of the Junior Order, United American Mechanics. In politics he is an independent, while his religious affiliations are with the Protestant Episcopal Church. Devoting practically all of his time to his business and to his family, Mr. Main has comparatively few outside interests, his chief hobby being radio and everything pertaining to it.
Mr. Main married, in 1922, Katherine Ordway, a native of New York City. Mr. and Mrs. Main have no children and make their home at Wakefield.

redpajama_set_name: RedPajamaCommonCrawl


Q: When can you start using minoxidil after a hair transplant?
A: Every clinic is different in their post op instructions. Some say as early as 3 days post op and some will say wait for two weeks. It is usually best to wait until the grafted area has healed as the minoxidil could cause a burning or stinging sensation from its alcohol content. However minoxidil is beneficial for use for the grafts and the natural surrounding hairs. I've even read where one hair specialist stated that there was no benefit from Minoxidil usage after a transplant. In my opinion, I believe that based on what research has found with minoxidil is that is should be beneficial to a patient after a transplant. However, check with your hair specialist.
Q: Is possible to lose my old hairs after hair transplant? In other words, I have lost a significant amount of hair that was not transplanted and I am 2 weeks post op. What should I do?
A: It's too early to see if it was damage to the preexisting hairs. The most common cause at this time period post op is hair shock loss. This can happened with any hair surgery but more common in the scalp. Wait, don't panic! It will come back over time. You could add Minoxidil topically once per day to your morning rituals (on the affected area or all of the scalp). It may speed the come back of the shocked area. Three months is about the average time you see in the return of shock loss without any intervention..
Q: Can I use rogaine even though I have seborrheic dermatitis on my scalp? I've heard I could.
A: This is not medical advice but an opinion from my physician and myself. The short answer is Yes you can, but it is better to use first a shampoo with Ketoconazol that can be prescribed in a higher concentration to reduce the seborrhea dermatitis in combination with the Rogaine Foam preparation at 5%. It would be better if you saw a Dermatologist or your hair specialist to properly diagnose the dermatitis.
Q: How much does hairline lowering surgery cost?
A: How old are you? This is very important to see how conservative we need to be in making your surgical plan. We need to evaluate how dense you have the sides and back of the head to give you an approximate number of grafts that we need to transplant to give you the result that you are satisfied with and appropriate for your age? If you are 30 years or younger and not have tried medical therapies, then we would discourage you from having a hairline augmentation of any kind. You could trap yourself with further work and waste this precious resource. You need to consult with a reputable hair specialist.
Q: What can an 18 year old male do to avoid hair loss?
A: First of all thank you for asking this question at your young age. I wish more young people who are experiencing hair loss would ask now when they notice it instead of waiting. Secondly, you need to have a comprehensive consultation with a hair specialist. That way you both can come up with a long tern master plan to intervene now and see where you are later. This consultation would include a family history. You need to check all your family line father and mother too. Androgenetic alopecia or Male Pattern Baldness (MPB) is the most common cause for hair loss and it can skip generations. A wellness exam is good to have a baseline of blood lab work and to rule out something else besides the genetic role. When we see patients losing their hair at a very young age, it is almost mandatory to look at every possible cause.
Q: I think I may have had a Chemical Burn from my hair stylist. I am freaking out. What can I do??????? Please! I look like a Chernobyl accident.
A: Try to be calm! I can try to imagine how do you feel and will try to offer an opinion. My best an honest advice is to see a dermatologist or even better, a reputable hair specialist for an evaluation to check in detail the scalp problem at hand. I would be lying to you if we said it will come back. We and you do not know the severity of the chemical burn. Please go in as soon as possible.
Q: Hey I am having bad crusting 1 month post op of my FUE from a clinic in Turkey? It itches. What can I do?
A: I am pretty sure you are a little bit worried about touching the grafts and have not washed daily to remove them slowly over time. It is one month post op and you should already have a clean scalp. But don't fret. I am sure you are fine. First go take a shower and shampoo the crusty areas. You will need to leave the shampoo in for 10 minutes, tap the area with your fingers to trying to remove very gently the crust. And yes it could affect the growth if it stays there. Check with your hair specialist if the crusting hurts too much or is difficult to remove. Do not dig and pick at it.
Q: I had a hair transplant 9 months ago. When I bath or shower, I see a lot of hair left in the bath and shower tray.
A: You need to go in for an evaluation with your hair specialist. It could be anything. Please make an appointment.
More Q and A blogs coming soon. Thomas Ortiz, BS Clinical director answers these various questions with his physician's approval as opinions.


Google has just introduced a service called Google Play App Signing that allows you to store your app signing keys on Google's servers. This means your keys can't be lost or maliciously destroyed, as sometimes happens. And you won't have to worry about multiple apps using the same key by accident (which also happens).
The new service means you can hand those keys to Google so that you don't have to worry about where they are or who has them. Google will then sign your apps with your key for you after you sign them with your new "upload key" and push them to Play. I don't know if this will allow for developers to continue to self-distribute with their previous keys outside of the Play Store, but assuming your app keys don't need to be changed to join the service then you should be able to self-sign as needed. Granted, that removes the convenience.
Keep in mind, opting into Google Play App Signing is a permanent change, and you cannot withdraw or remove your keys from their servers. That might mean that you'd have to re-release any apps again separately if you decide to leave, so make sure it's something you want to do.
This is another good tool to have come out of I/O for developers. After all, I/O is a developer conference, and Google certainly hasn't forgotten who makes the apps these days. If you are interested, feel free to check out the new Google Play App Signing in greater detail at the source below.


Advanced technologies like artificial intelligence (AI) and virtual reality (VR) are struggling to provide unparalleled user experiences. Live streaming videos, VR games, and tele-tourism are gaining popularity, providing the average consumer with new ways of interaction and connection to the outside world. However, people don't want to live in a virtual space; they want to enjoy real-life experiences and benefit from deeper connections to satisfy their needs.
With the help of modern mobile technologies, telepresence platform Ubiatar has turned to Blockchain technology to develop UbiatarPlay, a new project that will take tele-tourism to the next level. The Ubiatar service already has the technology, software, and apps to help UbiatarPlay gain recognition. With over 1,000 active users, and currently available on the Android and iOS marketplace, Ubiatar puts ubiquity into the hands of the people.
UbiatarPlay will leverage the Ubiatar technology, including apps and software to become a human empowerment network for people around the world. The end goal is to go beyond social networking with Avatars and Usars who can connect live and settle on different types of services. For example, an unemployed person in Singapore can become an Avatar, connect with an Usar (someone in New York City) and schedule a live tour of Singapore's hotspots.
In exchange, the Avatar receives financial compensation in UbiatarPlay tokens, whereas the Usar enjoys the unique user experience of visiting Singapore without being physically present. To ensure complete transactional transparency, UbiatarPlay will leverage smart contracts on Ethereum. All payments between users, regardless of their location, will be made in UbiatarPlay tokens. By cutting out the middleman, people won't have to deal with high exchange fees or banks to receive compensation for a provided service. Furthermore, transactions are made instantly into a user's Ubiatar wallet.
UbiatarPlay will use the powerful technology currently running on the Ubiatar service. The new project leverages the expertise of skilled developers and programmers to develop a peer-to-peer marketplace where users can reap the benefits of telepresence at the global scale through the Graphical User Interface (GUI).
Simple, yet extremely powerful, rather than write or ask for something by voice, Usars will click on the icons to elicit a desired action from Avatars. There is no need to guide the Avatar because they can understand simple iconic commands. By bringing the "Macintosh GUI" into the telepresence field, the team at Ubiatar has perfected it over the years to provide seamless interaction with just a few clicks.
To make sure the ubiquitous experience Usars get from Avatars is the real deal, UbiatarPlay technology relies on bi-dimensional video streaming via an Avatar's phone camera. To bring the experience to the next level, users can also attach VR systems to their phones to provide and enjoy 360-degree experiences and be able to truly feel that they are in a city or country located thousands of miles away from home without being physically present.
Blockchain technology comes into play to ensure all transactions on the UbiatarPlay platform are transparent, secure, and fast. Both Avatars and Usars will transact in UAC tokens, thus removing the need to handle high exchange fees when interacting with people located at the other end of the world.
The UbiatarPlay is scheduled for May 9. With an existing MVP live and running, upon successful completion of the public token sale, the team will focus on developing its VR system and perfecting the UbiatarPlay platform.


Our Stronghold for Nature, Exploring our Forest, Revealing our Past, Celebrating our Forest, Securing our Future.
Each of these themes has a range of projects that uncover, share and celebrate our heritage. There are plenty of opportunities for you to get involved in learning about and protecting our special forest, so we hope you'll come and join us!


We purchased a grain mill and learned how to grind our own wheat. We bought one that wasn't electric, and while it would be great with a grid down scenario, it just takes way too long to make this a part of our routine. Next year we hope to buy an electric one. We made muffins with our grain, and the freshness and the way it filled you up for the entire morning was wonderful. I want this to be a part of my routine, and truth is, I won't grind the wheat by hand unless I was in a grid down situation. But it does feel really great to know we have it!
I started a food storage business, Thrive Life. It was not an easy decision for me, in the past I have kept my food storage to myself, but because I believe preparation must be a way of life and something we all need to consider doing, I decided to be open about it, and sell the products. The surprise for me was how the quality of the food is so much better than what I purchase at the store! We are now incorporating the freeze-dried foods into our everyday pantry and finding it not only cost-effective but convenient.
The disaster in the Philippines was a wake-up call for me to realize that we had neglected water preparation. We live close to a river, so while we do have a water source, it would need to be purified. I bought us a water purifier and gave it a try with the water from the river. I must say, it was good! I just purchased a 55 gallon water system as well. It is on sale this month, for $87.89, it is a great deal! (The sale continues through Jan. 20, I need to put this order in through the back office for the sale price, so leave me a message if you are interested.) I also learned a lesson on storing water jugs on their side…we had a leak in one and it caused significant damage in our pantry. Now I will store the water in our shed. If it leaks, you've got a problem, especially if you have laminate flooring.
This year I hope to get my daughter and her family set up with their prepared pantry. Money is very limited for them, but there are some things I believe anyone can do on a limited budget.
I want to put together some jar meals, and have just ordered a book Meals in a Jar to help me get started. One I learned from having our kitchen torn apart, there are times when cooking is just too hard. I can see how in a disaster, this would be the case. For these meals, I simply add water and heat them up. I am looking forward to trying these recipes!
I also want to share more of these ideas with you here on the blog. I appreciate your feedback and comments, this helps me to know if what I am sharing is a help to you.
On a rainy cold day, just before Christmas, I saw a young woman by the side of the road holding a sign that said, Out of money, Out of gas, please help me. I was surprised, we live in a suburban setting, and while we don't see this often, we have been seeing it more and more. Truthfully, I rarely give to people on the side of road like this, but there was something about this young woman that I just couldn't shake. I had a gift card we received from the union to a local grocery store and so I stopped and she ran over, and I gave it to her. In her face I saw three things… relief, shame, and gratefulness. I cried all the way home. This is why we prepare, this is why we look forward, this is why we look back on what we have accomplished. And everyday, we pause and we give thanks.
Now tell me, what have you been accomplished this past year, and what are you plans for 2014?


Savvy consumers seek out the best deals.
Most consumers probably are unaware that when they don't complete checkout after adding items to an online shopping cart, they're becoming part of a statistic. It's called "shopping cart abandonment," and the trend is rising.
The shopping cart abandonment rate reportedly reached a peak average of 72 percent in 2011, with the highest abandonment rate of 89.2 percent reported on Nov. 20. The rate was 71 percent in 2010, and 68 percent in 2009.
Market analyst Charles Nicholls of SeeWhy believes one reason for this trend is the "deal seeking behavior" of online buyers.
"More and more consumers are becoming sophisticated Internet shoppers, aware of simple techniques to seek out the best deal and of TV-advertised price comparison websites," he notes.
Abandonment rates typically rise during the last quarter of the year because merchants offer new products and holiday promotions meant to entice buyers. Meanwhile, buyers are shopping around and tend to abandon their carts if a better deal is found elsewhere.
Online marketers should try to offer simpler checkout processes, better security and more payment options, as well as promotional deals for consumers, Nicholls advises. As much as 47 percent of online buyers reportedly will only buy discounted products, and 73 percent "rate unconditional free shipping as a critical feature," he notes.
During the end-of-year season, savvy buyers expect promotional offers and may abandon their shopping carts if the deals don't stack up.
In other words, those with the most online-shopping experience are more likely to abandon their shopping carts.
I do this a lot because sometimes it is the only way to get to the bottom line price including the shipping charges. Amazing, but there are some sites out there that expect payment upfront and then will tell you the shipping and handling amount after the fact. I don't even consider shopping at those places!
I agree with Cheryl. There are many times when the only way to find out how much shipping is, is to add the item to the cart. Some times it's the only way to even see the price of the item. Maybe some up front info would be better to avoid this situation. They seem to want all your info before they give you theirs.


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Stockbridge Bowl Association Achievements
History, Geography & Facts
Restoring Stockbridge Bowl
About Save Stockbridge Bowl Campaign
Home/Restoring Stockbridge Bowl – Plan B/Frequently Asked Questions
Frequently Asked QuestionsSBA Admin2019-03-19T14:16:47-04:00
The Board of Directors of the Stockbridge Bowl Association (SBA) thanks you for your concern about the health of Stockbridge Bowl. We invite you to join us in protecting our shared resource, now and for the future, as part of our Save Stockbridge Bowl Capital Campaign.
For more than half a century, invasive weeds have been multiplying near the shore and sediment has built up continuously. Inevitably, the sediment is getting near the surface in critical parts of the lake and "Stockbridge Bowl" is seriously in danger of becoming "Stockbridge Bog." This concern led the Board of Selectmen of the Town of Stockbridge to state in one Annual Report, "The future health of the lake is hanging in the balance. If we don't address these issues soon, the lake will decline and die."
Who maintains the Bowl?
The Commonwealth of Massachusetts owns all great ponds and lakes in the state, but the municipalities bordering those lakes are charged with protecting them. Because the Town of Stockbridge is responsible for maintaining the Bowl, the SBA turns the money we raise to restore the Bowl over to the Town. Once voters approve expenditures at Town Meeting, it is the Town that hires the consulting engineers and the contractors to undertake the work. This is the procedure recently followed in constructing the 150-foot long diversion drain that allows water to bypass a Town sewer and three liquid gas lines that were choking the outlet of the Bowl.
What needs fixing now?
Engineers have measured locations where sediment has piled up most deeply. These are where sediment needs to be removed to enable a 5.5-foot winter drawdown of the lake to freeze and kill the roots of invasive Eurasian Watermilfoil, the weed that grows prolifically near the shore all around the lake. The engineers determined that 20,000 cubic yards of sediment must first be dredged from the outlet and behind the Island to permit the drawdown. Not only would that kill much of the Milfoil, it would also expose additional sediment that needs to be removed in front of the Town Beach and next to the causeway. The dredged sediment would be removed in a single winter and be trucked to meadows at Bullard Woods, owned by the SBA at the north end of the lake. When dredging is complete, the meadows would be smoothed, reseeded and fully restored.
The Town's engineers estimate that it will cost $2.7 million for the dredging phase of the Lake Management Plan. The SBA already has raised $2 million in grants, contributions and pledges, of which $597,000 came from the Mass. Department of Environmental Protection. Stockbridge residents have also approved grants for the project of more than $400,000 from town funds and state matching money under the Community Preservation Act. The SBA is actively continuing our Save Stockbridge Bowl Capital Campaign so that the Bowl restoration project – in planning for nearly 20 years – can be completed.
Restoring Stockbridge Bowl – Plan B
> SBA Newsletters
The Stockbridge Bowl Association
Copyright © Stockbridge Bowl Association| All Rights Reserved | The Stockbridge Bowl Association is a 501c3 not-for-profit organization. Donations and contributions to the SBA are tax-deductible to the full extent of the law. | Designed by Studio Two


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Odrava (in tedesco Kulsam) è un comune della Repubblica Ceca facente parte del distretto di Cheb, nella regione di Karlovy Vary.

Geografia fisica
Il villaggio sorge vicino al bacino dell'Ohře, sulla riva destra dell'omonimo fiume (Wondreb in tedesco).

I comuni limitrofi sono Ava e Sebenbach ad ovest, Nebanice, Chotikov, Vackovec, Hartousov, Hnevin e Pochlovice a nord, Hlinova, Rollessengrun, Thurn e Liboc ad est e Lipoltov, Scheibenreuth, Jesenice e Drenice a sud.

Storia
La prima citazione scritta del paese risale al 1370, ma si ipotizza che esso risalga al XIII secolo. Odrava è stato uno dei più antichi villaggi in quella zona. Sotto il regno di Ottocaro I di Boemia il villaggio passò alla regione della Boemia, ma nel 1322 tornò al distretto di Cheb. Diventa un comune autonomo solo nel XX secolo.

Economia
La fabbrica di argilla, la più importante di tutto il distretto, risale al 1864. Essa era fornita di 7 forni fusori, 10 ruote del vasaio, 6 stufe e 3 presse idrauliche e dà lavoro a 50 uomini e 20 donne. Ogni anno produce 400000 bottigliette di acqua minerale, 6000 tubi per condutture d'acqua, 100000 tubi midollari, 300000 mattoni e 100000 tegole. L'impianto è rimasto questo fino al 1875, quando è stato necessario procurare altre apparecchiature, probabilmente a causa della concorrenza di altre grandi fabbriche ceche. Nel 1886 la fabbrica fu chiusa.

Successivamente l'economia del villaggio si incentrò maggiormente sull'agricoltura. Infatti, grazie alla vicinanza del fiume omonimo, le terre del villaggio erano molto fertili.

Località ed attrazioni
Castello Mostov (1850); nel più breve tempo possibile, è prevista per la ricostruzione dell'edificio, che dovrebbe servire da negozio di ceramica, museo della ceramica, sala concerti e albergo.
Ponte di pietra lungo il fiume Odrava, costruito nel 1841
Torre, 637 metri s.l.m.
Piste ciclabili e campi per equitazione, golf e tennis
Museo
Teatro
Galleria fotografica
Campeggio
Casinò

Geografia antropica

Frazioni
Odrava
Dobroše
Mostov
Obilná
Potočiště

Società

Evoluzione demografica

Galleria d'immagini

Note

Altri progetti

Collegamenti esterni

Comuni del distretto di Cheb

redpajama_set_name: RedPajamaWikipedia


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Tuesday was Australia's hottest day ever
Australia has sweltered through its hottest day on record, but that milestone could fall again before the week is out.
A stifling hot air mass and blistering heatwave conditions scorched central Australia as temperatures soared beyond 45C on Tuesday.
The air mass is so extensive, the average national temperature climbed to 40.9C on Tuesday, eclipsing the record of 40.3C set in January 2013.
The heatwave will only continue to intensify, with southern and central Australia experiencing temperatures up to 16C above average by Friday.
The Queensland outback town of Birdsville was the hottest place in the country on Wednesday, reaching a peak of 47.7C, not far ahead of Wudinna airport in South Australia which reached 47.3C.
Most of Queensland's interior is expected to reach at least 40C by the weekend, with temperatures up to 12C above average in some towns.
Western Australia's peak on Wednesday reached 46.9C in Mandora, south of Broome and the Northern Territory's hottest was Kintore, around 500km west of Alice Springs, at 46.4C.
SE Qld hit by almost 16,000 power outages
Severe storms hit South East Queensland again
Hero neighbours rescue woman from Melbourne floodwaters
Western NSW temperatures surged past 40C, with the mercury tipped to climb even higher across the state in coming days.
Melbourne reached 39.7C just before 3pm on Wednesday, while the state's northwest is expecting to crack the mid-40s.
Canberra was a relatively mild 36.7C and temperatures were a cooler 31C in Tasmania.
The heatwave has added to dangerous fire conditions with warnings current for most of the country and more than 170 fires burning in NSW and Queensland.
Australia's far southern coast is expected to see slightly cooler conditions on Thursday, but dangerous heat conditions will continue to trouble firefighters.
Strengthening winds will create very high to extreme fire dangers in northeast Victoria, NSW and the ACT with smoke becoming a significant issue.
"This combined with severe to extreme heatwave conditions to have a large impact on people's health," meteorologist Diana Eadie said.
The National Asthma Council has also warned the extreme heat could affect people's breathing and trigger symptoms.
The cold front will move over WA on Thursday and push further east late on Friday after another hot day in Adelaide, Melbourne and Hobart.
Auto News: Kia's Australian Open Fleet hits the road - caradvice.com.au


You found long and enduring adoration in your perfect partner and have been hitched for a significant number of years, with the sentiment of profound association with a long haul accomplice – advanced to empower you to stay with a mate at any rate sufficiently long to raise a solitary tyke through outset together as a group—albeit a considerable lot of us stay together any longer, and appreciate the advantages of existence with an accomplice notwithstanding when there is no objective to have kids and you anticipate that this connection should keep going forever.
And after that, BOOM your perfect partner passes on, presently you are single, alone, and lamenting. There is no "right" approach to feel subsequent to losing your mate. Such huge numbers of factors add to your response, including to what extent and cheerful your marriage was, the manner by which your life partner passed on, how old your kids are in the event that you have them, and how dependent you were on each other.
You may feel numb, stunned, despondent, or on edge. You may feel remorseful for being the person who is as yet alive or assuaged that your life partner is never again enduring in the event that the person in question was sick for quite a while. You may even feel furious at your companion for abandoning you. You may cry a great deal, or you may not.
However, the million dollar question is! So, how soon is too soon?
While the recently separated or divorced are allowed to take the field again when they like, the bereft must explore religious, family and community rules that govern this matter, and they change. All things considered, it's certainly up to you. What's more, it will be; everything relies upon the person. There are different variables that can affect the preparation to "proceed onward." Once you believe you're prepared, generally when you begin to question the fact if you are prepared or not, go and test it out.
Statistics show that men date sooner and remarry more rapidly than ladies do, and there is measurable legitimacy in this. The normal time period for single men who remarry is around two-three years while for widows; it's three to five years. In any case, having kids or not, being more youthful or more established and your general condition of flexibility despite disaster plays into this also.
At the point when a mate is on their debilitated bed, some may let you know, don't weep for me yet discover somebody after I am no more. It's so difficult to accept. Yet, the reality of the situation, is, yes you can respect their demand, as they are giving you their approval to go and observe love for you to be upbeat once more.
I was asking this same question, and a friend of mine revealed to me that a wife had malignant cancer and was in the hospital, a woman went to visit her friend more than once on the same ward whose bed was beside the wife with cancer. You know when people go to visit patients they welcome different patients.
Along these lines, the visitor would converse with the wife, and one day the wife disclosed to her, " I like you, you are exceptionally caring and I see how you care for your friend, and in the event that I die I would want you to take care of my husband". She knew the spouse as he was there most occasions when she visited her friend. Along these lines, the woman was shocked, and she wiped out the sick wife striking proclamation.
It is obtuse, and it's very lonely to get back home to an empty home, an empty bed around evening time. It is such a morbid inclination to go on an excursion for work and realize that you have nobody to call to disclose to them that you arrived securely. It is a horrendous inclination to realize that you never again have a friend or family member to be worried about you. So, my question to you is why would you not proceed onward with your life after your spouse passes on? When there are individuals out there on the planet who could bring you extraordinary euphoria and joy? Now, you need to make a stride back and ask yourself "What do I need for myself? You are your contemplations, what you think will show, so think positive. Try not to give humanity a chance to put impediments on your life. Not every person will be for you and that you should approve of during the time spent on fulfilling your needs. So, happy hunting!


Choose A New Voice For Your Google Home....!
Thanks for your Likes & Subs! Full written review: TechGadgetsCanada.com Smart digital assistants are growing in popularity, big time. In Canada, there's really only two choices right now: Amazon Echo and Google Home. Each brand has different versions of their assistants, usually a full size and a mini.At their core, these smart speakers/digital assistants operate largely the same no matter which size you choose, so we'll focus this comparison on the base models; the full size Echo and Home. Google Home vs Amazon Echo: Who you talking to? Human or Robot? Each of the home devices has a different Assistant personality: Google has the Google Assistant, aka "Google". Amazon has Alexa. While both have female voices by default, Google Assistant has a more normal and natural sounding voice and delivery. Alexa's delivery is rather robotic and vaguely European even though I've selected Canadian English as her voice. Will one of those voices please you more than the other? Maybe. At the moment, my preference is for Google Home. While you have some choice for accent changes with Alexa, for the moment you don't with Google Home. Here's how to do it: To change your Google assistant's voice: Go to Google Home -Hamburger Menu- Settings or More Settings- Preferences- Assistant Voice. ✅ Voice WINNER: Google Home So how do you know which of these smart speakers is for you? Here are some of the other choices you'll have to make when deciding between Google Home and Amazon Echo. Do Google Home and Amazon Echo do the same tasks? Yes. By and large these devices perform all the same basic functions: playing music (Google Play Music, Amazon Music, Spotify) answering questions setting timers controlling Smart Home gadgets like lights, thermostats, door locks and more telling you about your day; see more below getting news, weather and traffic info making conversions: money, measurements, time make phone calls look up phone numbers (and call them) add items to to-do lists and create reminders make calendar appointments; depending on your OS, see below Want to know how to talk to Google Home and Philips Hue smart lights? Google Home vs Amazon Echo: Are you Apple or Android? Both Amazon Echo and and Google Home are compatible with many different smartphone platforms, including Android, Apple and more.google home speaker assistant what why how to Google Home is definitely made to work closely with its own Pixel line of smartphones, and other Android phones and you may enjoy a more seamless experience if this is the platform you've got. As far as Amazon Echo goes, the device may seem to be geared towards the Android world, but again, there's a lot you can do with your Apple iPhone. Can Alexa link to Apple Calendar? With Echo you *can* add your calendar to Alexa's brain. She'll integrate with Apple Calendar/iCloud, Microsoft Outlook and Office, plus the G-suite of apps and functions from Google. Want to hear about your day or list of appointments from your Apple calendar? Alexa will tell you. ✅ Compatibility WINNER: Amazon Echo Google Home vs Amazon Echo: Smart home control Both Home and Echo can control your smart home devices, and the list is pretty much universal, though if you're wondering about controlling a specific smart home toy, it's a good idea to check it. But when it comes to door locks, lights and thermostats, they're both pretty much the same. If I noticed any difference its that it's much much easier to add a single new device to Echo. For example, when I wanted to add a single new LIFX light, with Google Home I need to uninstall and reinstall the whole LIFX lineup of several bulbs, whereas with Echo, I just add the one light and it works great. Google Home vs Amazon Alexa: Price Amazon Echo costs $129 CAD. Google Home is $179CAD. Given that Echo has more versatility and functionality, and it costs less, this battle goes to Alexa. ✅ Price WINNER: Amazon Echo So who's best? Amazon Echo or Google Home? All things considered, Amazon Echo does more and is more versatile, speaking to different platforms and devices (not just Android) and it costs less. If you're looking for a smart home digital assistant and speaker to start out with Amazon Echo is the winner. ✅ OVERALL WINNER: Amazon Echo Do you agree or disagree? Have any questions about what these devices can do? Ask in comments .
This video shows you how to change your Google Home voice using the Google Home app for iPhone, iPad and Android devices. See more videos by Max here: https://www.youtube.com/c/MaxDalton Video Transcript: Google Home speakers are a great way to build out your smart home. They feature good sound, and a wide variety of devices and services are integrated into Google Assistant. One aspect of Google Home devices that not everyone likes is the default voice. Thankfully, if you want to change your Google Home voice, you can do so quickly using the Google Home app for iPhone, iPad and Android devices. Now let's walk through the steps to change your Google Home voice. Step 1. Tap to launch the Google Home app on your iPhone, iPad or Android device. Your Google Home home screen appears. Step 2. Tap the "Settings" icon on your Google Home home screen. The Settings icon has an image of a gear on it. The Settings screen appears. Step 3. Scroll down to the bottom of the Settings screen, where you'll see a More Settings option. Tap "More Settings." You'll arrive at a screen with additional settings. Step 4. Tap "Assistant" in the row of options at the top of the menu. The Assistant options appear in the menu. Step 5. Tap "Assistant Voice" in the menu. The Assistant Voice menu appears. Step 6. Tap to place a check mark inside of a different color dot. As soon as you do this, a sample of your Google Assistant's voice will play. The voices will vary between male and female, and they will all have a different type of accent. For example, if you want to change your Google Home voice to be British, you could select the British Racing Green option. When you find the voice you want to change your Google Home voice to, leave that option checked and either tap to return to the Google Home home screen or close the Google Home app. The next time you interact with your Google Home, your device's voice will be changed.
Last month Google announced a lot of things at their Google I/O 2018 and among those are some new voices for your Google Assistant. So with a total of now 8 voices, lets see how easy it is to change that voice in your Google Home. ----------------------------------- Google Home Tutorials https://www.youtube.com/watch?v=9EqP1cynzv8&list=PL4KuquQflsLea0Yz-XgpPDzRYhnftXZPm ----------------------------------- ►Follow Me ! ➥ https://www.instagram.com/grosstechreview/ ➥ http://https://www.facebook.com/GrossTech/ ➥ https://twitter.com/GrossTechnology ----------------------------------- ►SUBSCRIBE YOUTUBE → https://www.youtube.com/c/Grosstechnology ----------------------------------- ►BACKGROUND MUSIC →Home DISCLAIMER: As an Amazon Associate I earn from qualifying purchases. This video and description contains affiliate links, which means that if you click on one of the product links, I'll receive a small commission. This helps support the channel and allows us to continue to make videos like this. Thank you for the support! List of gear: https://kit.com/GrossTechnology Tubebuddy link: https://www.tubebuddy.com/GrossTech MorningFame link: https://morningfa.me/invite/4khc1u54 Video Ranking Academy link: https://courses.seancannell.com/a/7802/tbJ3Fdfw In this video I go over just how easy it is to access and change that voice in your Google Home.
Full info: Blog.BestBuy.ca Follow me on Twitter & Instagram @ErinLYYC *Sorry about the cruddy audio on this one, friends. Still learning my new equipment.* You've no doubt heard about the popularity of voice-controlled technology. Did you know you can not only control speakers and lights by talking to them, you can also control your TV? It's becoming increasingly common to see voice-controlled TVs. You'll find a few options: voice control is built into the remote; you'll need to push a button to activate it; Examples: Apple TV voice control is built in to a smart TV. Just talk to it. Examples: Samsung Frame TV/smart TV and Android TV with Google Assistant voice control is added-on, usually via a streaming device like Google Chromecast, when connected to a smart home digital assistant such as Google Home. You may also see this with Apple TV and Siri via your iPhone too. What can you do with voice controlled TV? Depending which of the above scenarios you have, you can do different things. This article will focus on controlling a Google Chromecast device using Google Home. With voice control, you can ask the Google assistant to carry out several tasks like: Turn a Chromecast streamer on/off Play most streaming content by saying, "OK Google, Play (Show name) on (streaming service, i.e. Netflix, YouTube) on (name of Chromecast device, i.e "Living Room TV") Play music on the Chromecast to the TV Set up Google Home and Google Chromecast If you haven't already, plug in and set up/configure your Google Chromecast and/or your Google Home. Make sure you are logged into both devices using the Google Home app and ensure they are on the same Wi-Fi network. Link the Chromecast and the Home device To get Google Home to talk to Chromecast TV you need to link them in Google Home app. Open Google's Home app on your smartphone or tablet. You will likely see a 'card' on the main screen that asks you to "Link to your account" (If not, check that you can see both the Google Home and the Chromecast in the "Devices" tab.) Click Link to your account. Test the connection by saying, "OK Google, Play (TV Show name) on (streaming service, i.e. Netflix, YouTube) on (name of Chromecast device, i.e "Living Room TV") You should see the content start playing. If it's not quite configured, the Google Home app may prompt you to register or log in to some video services. How to log in to video services Google's Home app will likely prompt you to log in to things like YouTube and Nexflix, before you can use them but if it doesn't, open the app for the service you'll want to link, such as Netflix, and log in. At this point, return to the Home app and you should be linked. For the most part, the Google Home app will help you through this set up, even if you're using an iPhone, as I was.
Google Home Mini & Max assistant supports other languages like German, English, French, Japanese, and later this year 2018 will include Spanish, Italian, Portuguese and Korean. Learn more about Tech and how to use it, Subscribe to Tampatec here- https://www.youtube.com/tampatec * Follow Tampatec on Instagram for tech videos and tech giveaways -https://www.instagram.com/tampatec/ *Follow on G+ http://bit.ly/2ormuvN * I offer free tech support for my subs and followers *Some of these products I buy myself and some I receive for free to produce a Howto guide or / and review for companies but I DON'T get $ for positive product reviews. *My TV repair tools I use in my videos found here- http://amzn.to/2B7J1q9 the links support my tech channel Hi, I'm Paul an electronics tech from NYC area but live in Tampa bay area currently aka Tampatec, I post weekly usually on the weekends. My Howto tech channel is aimed for the novice tech or anyone that wants an easy to follow tech guide or wants to easily learn how to setup, use or fix their electronics. whether it's LED LCD TV, tablets, phone repair, home automation setup, Google Home, Chromecast guide, Echo dot tutorials, just to name a few. I'm here to help for free, just subscribe. *DISCLAIMER When working on electronics or installing, act at your own risk. follow safety manuals and / or NEC safety codes, unplug power, please watch other howto videos for a better understanding. Tampatec is not responsible for damages or misunderstanding info in the videos. I believe in this famous quote- give a person a fish they eat for a day but teach that person to fish and they will eat for a life time. Thanks for watching and your support!
Guide on how to create this: Product: Smarter iKettle version 3.0 Automation tools used: Smart Voice assist (either): Google Home Amazon Alexa Samsung Smarttings ActionTiles Options: Option 1: Using IFTTT and Google Home or Amazon Alexa: *. Set up your account with iKettle, follow the getting started guide that came with the smart kettle. *. Set up your account with IFTTT.com. *. Click on your user name in the top right and choose "Services" *. In the search bar search and add "Smarter" service (smarter is the company that made the iKettle). *. Follow the prompts to add your account. *. Do the same steps for either google assist or amazon alexa based on what you have/want. *. Go to IFTTT Applet page. (on the top left) *. Click on add a new Applet. *. Click on IF THIS and choose either google assist or amazon alexa as the service you like. *. Choose "Say a simple Phrase". *. Choose what you want to say, as an example in the video i choouse "Make me coffee". *. Click "Create Trigger". *. Now click on "That". *. Search for smarter. *. Choose "Boil the kettle" and fill in the way you want it. *. Complete and create the applet. Option 2: Using SmartThings and ActionTiles. *. Go to https://graph.api.smartthings.com/ and log into your ST account. *. Go to My Devices *. Click on "New Device". *. Name: Kettle. *. Type: Momentary Botton Tile. *. ID: Kettle001 *. Add the new device. *. Open ST app on your phone and make sure the new device is added. *. Go to SmartApps on your ST mobile app and choose IFTTT (if you dont yet added IFTTT SmartApp, do so now). *. Click on IFTTT and add the newly created switch button. *. Go back to IFTT and create a new applet (same steps as above only this time the IF THIS should be SmartThings as the service and choose the new button as the switch). ActionTiles: *. In AT (ActionTiles), resync ST "Things" to find the new button switch. \ *. Go back in AT and add a new "Item". *. Add the new Switch. *. Move the tile where you want it. *. Done.
A powerful little helper. Get hands-free help in any room with Google Home Mini. It's powered by the Google Assistant. You can ask it questions. Tell it to do things. Its your own personal Google. Get answers from Google. Use your voice to quickly find information about the weather, news, sports and more. "Hey Google, will I need an umbrella today?" Manage your day from breakfast to bed. Get help with things like your schedule, commute, travel information and more.Google Home Mini can even help you find a lost iPhone or Android phone and make phone calls to any personal contact or business. "Hey Google, wake me up at 6:30 AM tomorrow." Play, pause, and rewind. Hands-free. Control your favorite music, movies and shows, using only your voice. "Hey Google, play some pop music." Control your smart home. Use your voice to control your compatible lights, thermostats and more. Google Home Mini works with more than 1,000 smart devices from over 150 popular brands. "Hey Google, turn off the bedroom lights." Fun for the whole family. Learn something new, play fun games and listen to stories with the Google Assistant. Want to tell the kids breakfast is ready? Broadcast a message to other Google Home devices around the house. Hey Google, lets play a game. Your child can sign in to Google Home devices with their Google Account managed with Family Link. Your voice. Your info. Get personalized responses for everyone at home with voice recognition. "Hey Google, when is my first meeting tomorrow?" Choose between different colors. Google Home Mini comes in two unique colors, Chalk and Charcoal. Select the right color that matches your room and personal style. Learn New Technologies -- Visit Our Channel -- https://www.youtube.com/tutorialspoint2016 1) CodeIgniter Mini Project Tutorial in Hindi/Urdu Using Angular JS & Boostrap 3 : https://goo.gl/FSwsJ4 2) Codeigniter Tutorial for Beginners Step by Step in Hindi : https://goo.gl/bxG6gJ 3) PDO-OOP-PHP-CRUD-with-Bootstrap : https://goo.gl/q1KRTn 4) AngularJS Tutorial for Beginners (For Absolute Beginners ) : https://goo.gl/Hak26N 5) JSON Tutorial for Beginner : https://goo.gl/ryPu35 6) Git and GitHub Training In Hindi : https://goo.gl/HpToSd 7) Java Tutorial For Beginners (Step by Step tutorial) : https://goo.gl/QxJ68x 8) PHP Tutorial for Beginners (For Absolute Beginners) : https://goo.gl/S3urkU 9) OOPS Tutorial for Beginners in PHP : https://goo.gl/k2kJDM 10) Bootstrap Tutorial for Beginners : https://goo.gl/NGFDse 11) Magic Methods in PHP Tutorial : https://goo.gl/BdYLfJ 12) Build a PHP MVC Application : https://goo.gl/yqDCkb 13) Whats new in PHP 7 | (Introduction) : https://goo.gl/rTTKkr 14) CRUD with PHP and MySQLi Tutorial : https://goo.gl/yyjWD7 15) Technology Tips and Tricks : https://goo.gl/MpR952 16) MongoDB Tutorial for Beginners (Hindi) : https://goo.gl/NZXD1u Any questions or suggestions you may have, let me know in the comments below and I will try to reply as soon as I can. You can connect with us in social Media :- Youtube : https://www.youtube.com/tutorialspoint2016 Twitter: https://twitter.com/tutorialspoint2 facebook : https://www.facebook.com/tutorialspoint2016 Instragram : https://www.instagram.com/tutorialspoint2016 Google plus : https://plus.google.com/+TutorialsPoint2016 Blogger : http://tutorialspoint2016.blogspot.in/ Don't forget to hit the Subscribe & Like button!


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Wojny rzymsko-partyjskie – szereg wojen toczących się między Republiką Rzymską, a później Cesarstwem Rzymskim i Królestwem Partów od I wieku p.n.e. do III wieku n.e.

Wyprawa Krassusa
Zbrojne konflikty między Rzymem i Partami zaczęły się w roku 53 p.n.e., kiedy do kontrolowanej przez Partów Mezopotamii wkroczyły rzymskie legiony dowodzone przez wodza Krassusa. W Iranie toczyła się od ok. 58 p.n.e. wojna między królem Elymaidy Orodesem a jego bratem, królem Medii Atropatene Mitrydatesem o tron Króla Królów. Krassus miał nadzieję, że wmiesza się w spór lub skorzysta z zaangażowania braci w walce między sobą i zagarnie bogatą Mezopotamię, a może nawet na trwałe uzależni potężne i rozległe państwo partyjskie od siebie i Rzymu.

Krassus zawarł sojusze z nieposłusznymi wasalami Partów: królem Armenii Artawazdesem II i królem Osroene Abgarem II. 36 tysięcy Rzymian (35 tysięcy piechoty, tysiąc kawalerii) stoczyło pod miastem Carrhae bitwę z ponad 10 tysięcy partyjskich jeźdźców dowodzonych przez wasalnego księcia z klanu Suren, jednego z najznamienitszych partyjskich rodów. Dynastia Arsacydów była pochodzenia scytyjskiego i przyniosła do Iranu scytyjskie wzorce. Konni łucznicy, korzystając ze swojej szybkości zasypali rzymską piechotę gradem strzał, nie dając się jej zbliżyć. Rzymskie wojska poszły w rozsypkę, jedynie 10 tysięcy legionistów zdołało uniknąć śmierci lub niewoli. Zdobytych jeńców i sztandary legionowe Partowie zwrócili Rzymowi dopiero po 33 latach.

Syn Krassusa zginął w trakcie bitwy, sam Krassus zginął po bitwie podczas negocjacji z Partami z rąk pomniejszego dowódcy, głowę prokonsula zabrano na dwór Orodesa, gdzie posłużyła jako rekwizyt w inscenizacji "Bachantek" Eurypidesa. Wielu legionistów wziętych w niewolę zapędzono w głąb Iranu do przymusowej pracy, jak niegdyś czyniono z żołnierzami Seleukidów. Sam Orodes bez trudu odparł atak Armeńczyków i przymusił Artawazdesa II do uznania swojej supremacji, Abgara Partowie zabili. Partowie nie wykorzystali zwycięstwa i nie zaatakowali imperium rzymskiego od razu, bowiem Orodes przeprowadzał czystki w swoim państwie i zabił m.in. Surenę, autora wielkiego sukcesu.

Najazdy Partów na Syrię
W dwa lata po tym zwycięstwie Partowie uderzyli na ziemie Imperium Romanum. Orodes wysłał wojska pod wodzą swego syna, współrządcę Pakorusa. Kwestor Krassusa, Gajusz Kasjusz Longinus (przyszły zabójca Cezara) zebrał to, co zostało z armii swojego przełożonego i odparł atak młodego króla. Z dalszych działań musiał Pakorus zrezygnować, bowiem ojciec, nie ufający nawet własnemu synowi (nic dziwnego: sam pozbawił życia swego ojca, Fraatesa) wezwał go do Ktezyfontu. W 46 p.n.e. Pakorus powrócił do Syrii i wmieszał się w walki między stronikami Cezara a zwolennikami Pompejusza. Późna pora roku zniechęciła go jednak i zawrócił. Większe sukcesy odnieśli w roku 40 p.n.e. W Palestynie trwał konflikt sukcesyjny między przedstawicielami dynastii Machabeuszy: Janem Hirkanem II i Antygonem. Pakorus wtargnął do Palestyny, osadził Antygona na tronie w Jerozolimie a Hirkana oddał pod straż satrapy Górnej Mezopotamii. Tymczasem Kwintus Labienus, syn legata Cezara Tytusa Labienusa, rzymski dowódca i stronnik zabójców Cezara, którego niegdyś, podczas wojny z Markiem Antoniuszem Kasjusz wysłał do Orodesa z prośbą o pomoc, na czele innej armii zajął Syrię i prawie całą Azję Mniejszą. Wydawało się, że Partowie spełnią swoje marzenie o restytucji irańskiej władzy w dawnych zachodnich prowincjach imperium Achemenidów. Nieporozumienia między Pakorusem a Labienusem (który raczej wiązał z całą akcją własne, nie partyjskie ambicje) ułatwiło Rzymianom działanie. Marek Antoniusz posłał do Syrii swojego legata, Wentydiusza, który pokonał siły Pakorusa pod Amanus (39 p.n.e.) i Gindaros koło Antiochii (38 p.n.e.), wykorzystując błędy taktyczne partyjskiego dowództwa. Tymczasem Labienusa wyparto z Anatolii. W 38 r. Pakorus zginął i Partowie wycofali się z ziem rzymskich.

Wyprawa Marka Antoniusza
Marek Antoniusz próbował zrealizować marzenia Juliusza Cezara i Kleopatry i podbić ziemie Partów, ale mimo zgromadzenia 100 tysięcy żołnierzy jego wyprawa zakończyła się porażką. W 36 p.n.e. Antoniusz ruszył do Iranu przez Armenię, omijając Mezopotamię. Liczył na zwycięstwo z pomocą zbuntowanych wasali Króla Królów Fraatesa IV: króla Armenii Artawazdesa II i króla Medii Atropatene Artawazdesa I. Klęska pod Fraaspą i kłopoty z zaopatrzeniem zmusiły go do wycofania się. Porażka nie przeszkodziła Antoniuszowi w ogłoszeniu w Aleksandrii (34 p.n.e.) Królem Królów Cezariona (syna Cezara i Kleopatry), Aleksandra Heliosa królem Armenii oraz wykrojeniu dzieciom Kleopatry i swoim królestwa w terytorium partyjskim i rzymskim. W 33 p.n.e. nastąpiła druga nieudana inwazja Antoniusza na imperium Partów.

Wojny rzymsko-partyjskie w I w. n.e.
W 20 p.n.e. w Armenii nastąpił przewrót: obalono sprzyjającego Partom Artaksesa II i wyniesiono na tron opowiadającego się za zbliżeniem z Rzymem i gotowego uznać zwierzchnictwo cesarza Tigranesa III. August posłał Tyberiusza, by koronował Tigranesa III w jego imieniu i wspierał partyjskiego uzurpatora, Tiridatesa I. Walka z nim związała ręce Królowi Królów Fraatesowi IV. Fraates nie mógł sobie pozwolić na walkę z Rzymianami i zwrócił sztandary legionowe Krassusa. August obnosił się tym "zwycięstwem", ale rozpoczęły się wielowiekowe walki rzymsko-irańskie o Armenię.

Cesarze dynastii julijsko-klaudyjskiej i flawijskiej prowadzili politykę umacniania obecności Rzymu na wschodzie poprzez aneksje wasalnych królestw nadgranicznych (Kapadocja, Kommagene, Mała Armenia, Paflagonia), co nie sprzyjało dobrym stosunkom z Partami dążącymi do odbudowy potęgi Achemenidów. W 18 r. n.e. zmarł Orodes, król Armenii z rodu Arsacydów (domu panującego w Iranie). Na owego króla wybrano Artaksesa III, a Tyberiusz zatwierdził wybór i uznał go za swojego lennika. Król Partów Artabanus II, zmagający się z trudościami wewnętrznymi, początkowo nie ingerował, jednak w 35 r. wysłał swojego syna, Arsakesa, aby zajął Armenię. Tyberiusz nasłał przeciw Partom króla Iberii Mitrydatesa, a ten wkroczył do Armenii i zachęcił Tiridatesa II, krewnego Artabanosa, do buntu. W Iranie wybuchła więc nowa wojna domowa, do której mieszał się namiestnik Syrii Witeliusz. Artabanus pokonał pretendenta, lecz Armenii nie odzyskał.

W 51 r. Mitrydatesa, władcę Iberii i Armenii, zamordowano. Do Armenii wkroczył namiestnik Kapadocji, a to wywołało reakcję Króla Królów Wologazesa I. Osadził on w Armenii swojego brata, Tiridatesa I. Neron wysłał Gnejusza Domicjusza Korbulona, by załatwił sprawę. Korbulon wysłał protest do Wologazesa, a ten początkowo ustąpił, jednak zbierał siły na walkę z Rzymem. To samo czynili Rzymianie (żadna ze stron nie była pewna przewagi). Tiridates zaatakował w 57 r. W pierwszej potyczce gdzieś nad Euftratem pokonał rzymskie siły i zajął Armenię. Unikał jednak walnej bitwy i pozwolił Korbulonowi wkroczyć w głąb swojego państwa. Wologazes, zajęty powstaniem w Hyrkanii i inwazją Kuszanów na Baktrię, nie mógł pomóc bratu. Korbulon spalił Artaksatę i w 59 r. ruszył na Tigranocertę. Rzymianie przywrócili do władzy Tigranesa VI. W 61 r. Tigranes VI najechał Mezopotamię – Partowie odpowiedzieli ponownym zajęciem Armenii i usunięciem go z tronu. W 62 r., gdy Wologazes oblegał Tigranesa w Tigranocercie, Korbulon zawarł rozejm z Wielkim Królem. Jeszcze w tym samym roku namiestnik Kapadocji Cesseniusz Petus wkroczył do Armenii, został jednak pokonany pod Randeją. Korbulon wdarł się do Mezopotamii, ale na wieść o klęsce Petusa zawrócił. Odnowiono rozejm: odtąd w Armenii panował Tiridates, brat króla Partów, ale jako wasal Nerona. W 66 r. Tigranes złożył hołd Neronowi. Ten kompromis, mimo późniejszych licznych wojen, okazał się trwały.

Wyprawa Trajana
W roku 113 naszej ery cesarz Trajan skierował przeciwko Partom całą potęgę Rzymu i do roku 115 opanował Armenię, Asyrię i Mezopotamię, zdobył Ktezyfon i Tygrysem dopłynął do Zatoki Perskiej. Wtedy rzymskie legiony mogły po raz pierwszy i ostatni zaznać kąpieli w wodach akwenu. Osadził na partyjskim tronie swojego protegowanego, a prawowity wielki król, Orodes, uciekł w głąb Iranu. Jednak zdobyte ziemie pozostały pod władzą Rzymu tylko przez chwilę, gdyż powstanie ludności mezopotamskiej zmusiło Trajana do wycofania się, a Hadrian, który został cesarzem w roku 117 nakazał opuścić zdobyte prowincje ze względu na zbyt wysokie koszty ich obrony i ciągłe zagrożenie.

Wojny rzymsko-partyjskie w II w. n.e.
Kolejna wojna między Rzymem a Partami wybuchła w roku 162, gdy partyjskie armie Wologazesa IV wdarły się do Azji Mniejszej, gdzie pokonały wojska namiestnika Kapadocji Sedacjusza Seweriana, i Syrii, gdzie zwyciężyły Rzymian pod Elegeją. Marek Aureliusz starał się zawrzeć kompromisowy układ, jednak Król Królów nie ustąpił. Już rok później rzymska armia dowodzona przez cesarza Lucjusza Werusa (a właściwie namiestnka Syrii Awidiusza Kasjusza i namiestnika Kapadocji Stacjusza Pryskusa) wkroczyła do Armenii i Mezopotamii pokonując Partów pod Dura Europos, a następnie paląc Artaksatę, Seleukię i Ktezyfont (w roku 165), a następnie ruszyła w głąb Wyżyny Irańskiej, gdzie została zatrzymana i zmuszona do odwrotu przez dżumę. Wracający do domu żołnierze zawlekli tę chorobę do Rzymu. W wyniku paktu zawartego w 166 r. z Wologazesem cesarz przejął od króla Partów zwierzchnictwo nad Osroene. Po 14 latach Partowie znów osadzili swojego przedstawiciela na tronie armeńskim.

Podczas wojny Septymiusza Sewera z Pesceniuszem Nigrem Wielki Król Wologazes V zgłosił gotowość poparcia tego drugiego, a król Osroene Abgar IX ogłosił niezależność. W 193/4 r. Sewer wkroczył do Mezopotamii, anektował Osroene zamieniając je w prowincję, a następnie wkroczył do Medii Adiabene i zawarł korzystny pokój z królem Partów. W 197 r. Wologazes zaatakował Nisibis. Cesarz powrócił na Wschód i ponownie zwyciężając Partów zajął Ktezyfont. Partowie musieli oddać Rzymianom dużą część Mezopotamii, jednakże rzymskie rządy nad Eufratem i Tygrysem skończyło się po śmierci Septymiusza Sewera.

Wyprawa Karakalli
Ostatnim cesarzem, który podjął wyprawę przeciw Partom był Karakalla, który, podobnie jak Trajan, marzył o powtórzeniu czynu Aleksandra Wielkiego. Kolejna wojna domowa w Iranie (Wologazes VI kontra Artabanus IV) i powolny rozkład imperium Partów miały dopomóc mu w odniesieniu zwycięstwa. W 216 r. cesarz anektował Armenię i wkroczył do Mezopotamii, a potem do Medii. Został zamordowany przez Marka Opiliusza Makrynusa, który zagarnął władzę. Makrynus stoczył nierozstrzygniętą bitwę pod Nisibis i celem zawarcia pokoju wypłacił Artabanusowi IV kontrybucję wojenną (200 milionów sesterców). Odnowiono traktat zawarty między Neronem a Wologazesem I, królem Armenii został Tiridates IV z rodu Arsacydów.

Ciągnące się wojny z Rzymem, niepokoje wewnętrzne i najazdy koczowników osłabiły Partów i zostali oni w połowie lat 20. III wieku obaleni przez perską dynastię Sasanidów (swoich wasali), którzy szybko zapragnęli rozszerzyć swoje imperium (więcej informacji na ten temat w artykule wojny rzymsko-sasanidzkie).

Najważniejsze bitwy między Rzymianami i Partami

Walki w latach 56-51 p.n.e.
 53 p.n.e. – bitwa pod Carrhae (klęska wyprawy Krassusa)

Walki w latach 40-36 p.n.e.
 40 p.n.e. – bitwa pod Apameą (zwycięstwo Partów dowodzonych przez Pakorusa I i Kwintusa Labienusa nad wojskami rzymskimi w Syrii)
 39 p.n.e. – bitwa w górach Amanu (Rzymianie dowodzeni przez Wentidiusza pobili Partów dowodzonych przez Farnapatesa, który poniósł śmierć)
 38 p.n.e. – bitwa pod górą Gindaros (Rzymianie pod wodzą Wentidiusza zwyciężyli Partów dowodzonych przez Pakorusa, który poniósł śmierć).
 36 p.n.e. – bitwa pod Fraaspą

Późniejsze walki
 Walki z Partami 58 – 63 r. n.e. (Wojna Korbulona)
 Walki z Partami 114-117 r. n.e. (Wojna Trajana)
 Walki z Partami 162-165 r. n.e. (Wojna Markusa Aureliusza)
 Walki z Partami 197-199 r. n.e. (Wojny Septimiusza Serverusa)
 Walki z Partami 215-217 r. n.e. (Wojna Karakalli)

Partyjsko-rzymskie
Wojny starożytnego Iranu


text:
In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space  as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somehow more sophisticated in that it uses linear algebra (or some functional analysis) more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.

Calculus on Euclidean space is also a local model of calculus on manifolds, a theory of functions on manifolds.

Basic notions

Functions in one real variable
This section is a brief review of function theory in one-variable calculus.

A real-valued function  is continuous at  if it is approximately constant near ; i.e.,

In contrast, the function  is differentiable at  if it is approximately linear near ; i.e., there is some real number  such that

(For simplicity, suppose . Then the above means that  where  goes to 0 faster than h going to 0 and, in that sense,  behaves like .)

The number  depends on  and thus is denoted as . If  is differentiable on an open interval  and if  is a continuous function on , then  is called a C1 function. More generally,  is called a Ck function if its derivative  is Ck-1 function. Taylor's theorem states that a Ck function is precisely a function that can be approximated by a polynomial of degree k.

If  is a C1 function and  for some , then either  or ; i.e., either  is strictly increasing or strictly decreasing in some open interval containing a. In particular,  is bijective for some open interval  containing . The inverse function theorem then says that the inverse function  is differentiable on U with the derivatives: for

Derivative of a map and chain rule
For functions  defined in the plane or more generally on an Euclidean space , it is necessary to consider functions that are vector-valued or matrix-valued. It is also conceptually helpful to do this in an invariant manner (i.e., a coordinate-free way). Derivatives of such maps at a point are then vectors or linear maps, not real numbers.

Let  be a map from an open subset  of  to an open subset  of . Then the map  is said to be differentiable at a point  in  if there exists a (necessarily unique) linear transformation , called the derivative of  at , such that

where  is the application of the linear transformation  to . If  is differentiable at , then it is continuous at  since
 as .

As in the one-variable case, there is

This is proved exactly as for functions in one variable. Indeed, with the notation , we have:

Here, since  is differentiable at , the second term on the right goes to zero as . As for the first term, it can be written as:

Now, by the argument showing the continuity of  at , we see   is bounded. Also,  as  since  is continuous at . Hence, the first term also goes to zero as  by the differentiability of  at .

The map  as above is called continuously differentiable or  if it is differentiable on the domain and also the derivatives vary continuously; i.e.,  is continuous.

As a linear transformation,  is represented by an -matrix, called the Jacobian matrix  of  at  and we write it as:

Taking  to be ,  a real number and  the j-th standard basis element, we see that the differentiability of  at  implies:

where  denotes the i-th component of . That is, each component of  is differentiable at  in each variable with the derivative . In terms of Jacobian matrices, the chain rule says ; i.e., as ,

which is the form of the chain rule that is often stated.

A partial converse to the above holds. Namely, if the partial derivatives  are all defined and continuous, then  is continuously differentiable. This is a consequence of the mean value inequality:

(This version of mean value inequality follows from mean value inequality in  applied to the function , where the proof on mean value inequality is given.)

Indeed, let . We note that, if , then

For simplicity, assume  (the argument for the general case is similar). Then, by mean value inequality, with the operator norm ,

which implies  as required.

Example: Let  be the set of all invertible real square matrices of size n. Note  can be identified as an open subset of  with coordinates . Consider the function  = the inverse matrix of  defined on . To guess its derivatives, assume  is differentiable and consider the curve  where  means the matrix exponential of . By the chain rule applied to , we have:
.
Taking , we get:
.
Now, we then have:

Since the operator norm is equivalent to the Euclidean norm on  (any norms are equivalent to each other), this implies  is differentiable. Finally, from the formula for , we see the partial derivatives of  are smooth (infinitely differentiable); whence,  is smooth too.

Higher derivatives and Taylor formula
If  is differentiable where  is an open subset, then the derivatives determine the map , where  stands for homomorphisms between vector spaces; i.e., linear maps. If  is differentiable, then . Here, the codomain of  can be identified with the space of bilinear maps by:

where  and  is bijective with the inverse  given by . In general,  is a map from  to the space of -multilinear maps .

Just as  is represented by a matrix (Jacobian matrix), when  (a bilinear map is a bilinear form), the bilinear form  is represented by a matrix called the Hessian matrix of  at ; namely, the square matrix  of size  such that , where the paring refers to an inner product of , and  is none other than the Jacobian matrix of . The -th entry of  is thus given explicitly as .

Moreover, if  exists and is continuous, then the matrix  is symmetric, the fact known as the symmetry of second derivatives. This is seen using the mean value inequality. For vectors  in , using mean value inequality twice, we have:

which says

Since the right-hand side is symmetric in , so is the left-hand side: . By induction, if  is , then the k-multilinear map  is symmetric; i.e., the order of taking partial derivatives does not matter.

As in the case of one variable, the Taylor series expansion can then be proved by integration by parts:

Taylor's formula has an effect of dividing a function by variables, which can be illustrated by the next typical theoretical use of the formula.

Example: Let  be a linear map between the vector space  of smooth functions on  with rapidly decreasing derivatives; i.e.,  for any multi-index . (The space  is called a Schwartz space.) For each  in , Taylor's formula implies we can write:

with , where  is a smooth function with compact support and . Now, assume  commutes with coordinates; i.e., . Then
.
Evaluating the above at , we get  In other words,  is a multiplication by some function ; i.e., . Now, assume further that  commutes with partial differentiations. We then easily see that  is a constant;  is a multiplication by a constant.

(Aside: the above discussion almost proves the Fourier inversion formula. Indeed, let  be the Fourier transform and the reflection; i.e., . Then, dealing directly with the integral that is involved, one can see  commutes with coordinates and partial differentiations; hence,  is a multiplication by a constant. This is almost a proof since one still has to compute this constant.)

A partial converse to the Taylor formula also holds; see Borel's lemma and Whitney extension theorem.

Inverse function theorem and submersion theorem

A -map with the -inverse is called a -diffeomorphism. Thus, the theorem says that, for a map  satisfying the hypothesis at a point ,  is a diffeomorphism near  For a proof, see .

The implicit function theorem says: given a map , if ,  is  in a neighborhood of  and the derivative of  at  is invertible, then there exists a differentiable map  for some neighborhoods  of  such that . The theorem follows from the inverse function theorem; see .

Another consequence is the submersion theorem.

Integrable functions on Euclidean spaces
A partition of an interval  is a finite sequence . A partition  of a rectangle  (product of intervals) in  then consists of partitions of the sides of ; i.e., if , then  consists of  such that  is a partition of .

Given a function  on , we then define the upper Riemann sum of it as:

where
 is a partition element of ; i.e.,  when  is a partition of .
The volume  of  is the usual Euclidean volume; i.e., .
The lower Riemann sum  of  is then defined by replacing  by . Finally, the function  is called integrable if it is bounded and . In that case, the common value is denoted as .

A subset of  is said to have measure zero if for each , there are some possibly infinitely many rectangles  whose union contains the set and

A key theorem is

The next theorem allows us to compute the integral of a function as the iteration of the integrals of the function in one-variables:

In particular, the order of integrations can be changed.

Finally, if  is a bounded open subset and  a function on , then we define  where  is a closed rectangle containing  and  is the characteristic function on ; i.e.,  if  and  if  provided  is integrable.

Surface integral
If a bounded surface  in  is parametrized by  with domain , then the surface integral of a measurable function  on  is defined and denoted as:

If  is vector-valued, then we define

where  is an outward unit normal vector to . Since , we have:

Vector analysis

Tangent vectors and vector fields
Let  be a differentiable curve. Then the tangent vector to the curve  at  is a vector  at the point  whose components are given as:
.

For example, if  is a helix, then the tangent vector at t is:

It corresponds to the intuition that the a point on the helix moves up in a constant speed.

If  is a differentiable curve or surface, then the tangent space to  at a point p is the set of all tangent vectors to the differentiable curves  with .

A vector field X is an assignment to each point p in M a tangent vector  to M at p such that the assignment varies smoothly.

Differential forms
The dual notion of a vector field is a differential form. Given an open subset  in , by definition, a differential 1-form (often just 1-form)  is an assignment to a point  in  a linear functional  on the tangent space  to  at  such that the assignment varies smoothly. For a (real or complex-valued) smooth function , define the 1-form  by: for a tangent vector  at ,

where  denotes the directional derivative of  in the direction  at . For example, if  is the -th coordinate function, then ; i.e.,  are the dual basis to the standard basis on . Then every differential 1-form  can be written uniquely as

for some smooth functions  on  (since, for every point , the linear functional  is a unique linear combination of  over real numbers). More generally, a differential k-form is an assignment to a point  in  a vector  in the -th exterior power  of the dual space  of  such that the assignment varies smoothly. In particular, a 0-form is the same as a smooth function. Also, any -form  can be written uniquely as:

for some smooth functions .

Like a smooth function, we can differentiate and integrate differential forms. If  is a smooth function, then  can be written as:

since, for , we have: . Note that, in the above expression, the left-hand side (whence the right-hand side) is independent of coordinates ; this property is called the invariance of differential.

The operation  is called the exterior derivative and it extends to any differential forms inductively by the requirement (Leibniz rule)

where  are a p-form and a q-form.

The exterior derivative has the important property that ; that is, the exterior derivative  of a differential form  is zero. This property is a consequence of the symmetry of second derivatives (mixed partials are equal).

Boundary and orientation

A circle can be oriented clockwise or counterclockwise. Mathematically, we say that a subset  of  is oriented if there is a consistent choice of normal vectors to  that varies continuously. For example, a circle or, more generally, an n-sphere can be oriented; i.e., orientable. On the other hand, a Möbius strip (a surface obtained by identified by two opposite sides of the rectangle in a twisted way) cannot oriented: if we start with a normal vector and travel around the strip, the normal vector at end will point to the opposite direction.

The proposition is useful because it allows us to give an orientation by giving a volume form.

Integration of differential forms
If  is a differential n-form on an open subset M in  (any n-form is that form), then the integration of it over  with the standard orientation is defined as:

If M is given the orientation opposite to the standard one, then  is defined as the negative of the right-hand side.

Then we have the fundamental formula relating exterior derivative and integration:

Here is a sketch of proof of the formula. If  is a smooth function on  with compact support, then we have:

(since, by the fundamental theorem of calculus, the above can be evaluated on boundaries of the set containing the support.) On the other hand,

Let  approach the characteristic function on . Then the second term on the right goes to  while the first goes to , by the argument similar to proving the fundamental theorem of calculus.

The formula generalizes the fundamental theorem of calculus as well as Stokes' theorem in multivariable calculus. Indeed, if  is an interval and , then  and the formula says:
.
Similarly, if  is an oriented bounded surface in  and , then  and similarly for  and . Collecting the terms, we thus get:

Then, from the definition of the integration of , we have
where  is the vector-valued function and . Hence, Stokes' formula becomes

which is the usual form of the Stokes' theorem on surfaces. Green's theorem is also a special case of Stokes' formula.

Stokes' formula also yields a general version of Cauchy's integral formula. To state and prove it, for the complex variable  and the conjugate , let us introduce the operators

In these notations, a function  is holomorphic (complex-analytic) if and only if  (the Cauchy–Riemann equations).
Also, we have:

Let  be a punctured disk with center .
Since  is holomorphic on , We have:
.

By Stokes' formula,

Letting  we then get:

Winding numbers and Poincaré lemma

A differential form  is called closed if  and is called exact if  for some differential form  (often called a potential). Since , an exact form is closed. But the converse does not hold in general; there might be a non-exact closed form. A classic example of such a form is:
,
which is a differential form on . Suppose we switch to polar coordinates:  where . Then

This does not show that  is exact: the trouble is that  is not a well-defined continuous function on . Since any function  on  with  differ from  by constant, this means that  is not exact. The calculation, however, shows that  is exact, for example, on  since we can take  there.

There is a result (Poincaré lemma) that gives a condition that guarantees closed forms are exact. To state it, we need some notions from topology. Given two continuous maps  between subsets of  (or more generally topological spaces), a homotopy from  to  is a continuous function  such that  and . Intuitively, a homotopy is a continuous variation of one function to another. A loop in a set  is a curve whose starting point coincides with the end point; i.e.,  such that . Then a subset of  is called simply connected if every loop is homotopic to a constant function. A typical example of a simply connected set is a disk . Indeed, given a loop , we have the homotopy  from  to the constant function . A punctured disk, on the other hand, is not simply connected.

Geometry of curves and surfaces

Moving frame
Vector fields  on  are called a frame field if they are orthogonal to each other at each point; i.e.,  at each point. The basic example is the standard frame ; i.e.,  is a standard basis for each point  in . Another example is the cylindrical frame

For the study of the geometry of a curve, the important frame to use is a Frenet frame  on a unit-speed curve  given as:

The Gauss–Bonnet theorem
The Gauss–Bonnet theorem relates the topology of a surface and its geometry.

Calculus of variations

Method of Lagrange multiplier

The set  is usually called a constraint.

Example: Suppose we want to find the minimum distance between the circle  and the line . That means that we want to minimize the function , the square distance between a point  on the circle and a point  on the line, under the constraint . We have:

Since the Jacobian matrix of  has rank 2 everywhere on , the Lagrange multiplier gives:

If , then , not possible. Thus,  and

From this, it easily follows that  and . Hence, the minimum distance is  (as a minimum distance clearly exists).

Here is an application to linear algebra. Let  be a finite-dimensional real vector space and  a self-adjoint operator. We shall show  has a basis consisting of eigenvectors of  (i.e.,  is diagonalizable) by induction on the dimension of . Choosing a basis on  we can identify  and  is represented by the matrix . Consider the function , where the bracket means the inner product. Then . On the other hand, for , since  is compact,  attains a maximum or minimum at a point  in . Since , by Lagrange multiplier, we find a real number  such that  But that means . By inductive hypothesis, the self-adjoint operator ,  the orthogonal complement to , has a basis consisting of eigenvectors. Hence, we are done. .

Weak derivatives
Up to measure-zero sets, two functions can be determined to be equal or not by means of integration against other functions (called test functions). Namely, the following sometimes called the fundamental lemma of calculus of variations:

Given a continuous function , by the lemma, a continuously differentiable function  is such that  if and only if

for every . But, by integration by parts, the partial derivative on the left-hand side of  can be moved to that of ; i.e.,

where there is no boundary term since  has compact support. Now the key point is that this expression makes sense even if  is not necessarily differentiable and thus can be used to give sense to a derivative of such a function.

Note each locally integrable function  defines the linear functional  on  and, moreover, each locally integrable function can be identified with such linear functional, because of the early lemma. Hence, quite generally, if  is a linear functional on , then we define  to be the linear functional  where the bracket means . It is then called the weak derivative of  with respect to . If  is continuously differentiable, then the weak derivate of it coincides with the usual one; i.e., the linear functional  is the same as the linear functional determined by the usual partial derivative of  with respect to .  A usual derivative is often then called a classical derivative. When a linear functional on  is continuous with respect to a certain topology on , such a linear functional is called a distribution, an example of a generalized function.

A classic example of a weak derivative is that of the Heaviside function , the characteristic function on the interval . For every test function , we have:

Let  denote the linear functional , called the Dirac delta function (although not exactly a function). Then the above can be written as:

Cauchy's integral formula has a similar interpretation in terms of weak derivatives. For the complex variable , let . For a test function , if the disk  contains the support of , by Cauchy's integral formula, we have:

Since , this means:

or

In general, a generalized function is called a fundamental solution for a linear partial differential operator if the application of the operator to it is the Dirac delta. Hence, the above says  is the fundamental solution for the differential operator .

Hamilton–Jacobi theory

Calculus on manifolds

Definition of a manifold
This section requires some background in general topology.

A manifold is a Hausdorff topological space that is locally modeled by an Euclidean space. By definition, an atlas of a topological space  is a set of maps , called charts, such that
 are an open cover of ; i.e., each  is open and ,
 is a homeomorphism and
 is smooth; thus a diffeomorphism.
By definition, a manifold is a second-countable Hausdorff topological space with a maximal atlas (called a differentiable structure); "maximal" means that it is not contained in strictly larger atlas. The dimension of the manifold  is the dimension of the model Euclidean space ; namely,  and a manifold is called an n-manifold when it has dimension n. A function on a manifold  is said to be smooth if  is smooth on  for each chart  in the differentiable structure.

A manifold is paracompact; this has an implication that it admits a partition of unity subordinate to a given open cover.

If  is replaced by an upper half-space , then we get the notion of a manifold-with-boundary. The set of points that map to the boundary of  under charts is denoted by  and is called the boundary of . This boundary may not be the topological boundary of . Since the interior of  is diffeomorphic to , a manifold is a manifold-with-boundary with empty boundary.

The next theorem furnishes many examples of manifolds.

For example, for , the derivative  has rank one at every point  in . Hence, the n-sphere  is an n-manifold.

The theorem is proved as a corollary of the inverse function theorem.

Many familiar manifolds are subsets of . The next theoretically important result says that there is no other kind of manifolds. An immersion is a smooth map whose differential is injective. An embedding is an immersion that is homeomorphic (thus diffeomorphic) to the image.

The proof that a manifold can be embedded into  for some N is considerably easier and can be readily given here. It is known  that a manifold has a finite atlas . Let  be smooth functions such that  and  cover  (e.g., a partition of unity). Consider the map

It is easy to see that  is an injective immersion. It may not be an embedding. To fix that, we shall use:

where  is a smooth proper map. The existence of a smooth proper map is a consequence of a partition of unity. See  for the rest of the proof in the case of an immersion.

Nash's embedding theorem says that, if  is equipped with a Riemannian metric, then the embedding can be taken to be isometric with an  expense of increasing ; for this, see this T. Tao's blog.

Tubular neighborhood and transversality

A technically important result is:

This can be proved by putting a Riemannian metric on the manifold . Indeed, the choice of metric makes the normal bundle  a complementary bundle to ; i.e.,  is the direct sum of  and . Then, using the metric, we have the exponential map  for some neighborhood  of  in the normal bundle  to some neighborhood  of  in . The exponential map here may not be injective but it is possible to make it injective (thus diffeomorphic) by shrinking  (for now, see see ).

Integration on manifolds and distribution densities

The starting point for the topic of integration on manifolds is that there is no invariant way to integrate functions on manifolds. This may be obvious if we asked: what is an integration of functions on a finite-dimensional real vector space? (In contrast, there is an invariant way to do differentiation since, by definition, a manifold comes with a differentiable structure). There are several ways to introduce integration theory to manifolds:
Integrate differential forms.
Do integration against some measure.
Equip a manifold with a Riemannian metric and do integration against such a metric.

For example, if a manifold is embedded into an Euclidean space , then it acquires the Lebesgue measure restricting from the ambient Euclidean space and then the second approach works. The first approach is fine in many situations but it requires the manifold to be oriented (and there is a non-orientable manifold that is not pathological). The third approach generalizes and that gives rise to the notion of a density.

Generalizations

Extensions to infinite-dimensional normed spaces

The notions like differentiability extend to normed spaces.

See also
 Differential geometry of surfaces
 Integration along fibers
 Lusin's theorem
 Density on a manifold

Notes

Citations

References

 (revised 1990, Jones and Bartlett; reprinted 2014, World Scientific) [this text in particular discusses density]

Calculus