Question: Why will the resulting force lines of two positive point charges be like this:

![enter image description here](https://i.stack.imgur.com/3jnkx.jpg)

I would expect this:
![enter image description here](https://i.stack.imgur.com/KIltz.jpg)

Answer: 
Not only can but you should. Goldstein is a great book. Good mixture between math and ideas.
You can, certainly, it is a good book. But there are many other good books that you might also appreciate, with more advanced math. E.g. the "classics"

V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Vol. 60, Springer, 1989

R. Abraham and J.E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, 2008.

J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 1999.

Michael Spivak's new book on mechanics (Spivak, Physics for Mathematicians. Mechanics I, 2010) seems to be great, too. See <http://mathpop.com/mechanics1.htm>.