doctoral degree in physics, as it did with me. It’s physical underpinnings and mathematical grammar should be learned by the undergraduate between the sophomore and senior year, to be fully mastered during the first year and a half of graduate school where one should be learning the details of advanced mechanics, electrodynamics, quantum physics, thermodynamics and statistical physics, quantum electrodynamics, relativity, and possibly quantum fields with the power of a unified mathematical grammar and physical overview of where shit fits together and why it fits together. (Note—It doesn’t generally go backwards uniquely from postulating matrix representations back to (infinitesimal generator) realizations, back to differential equations, but this is nevertheless also a valid approach to investigate theoretical universes and extensions beyond the Standard Model.) There is, of course, more mathematics and physics beyond the unified outline expounded on in this work, but it ties back. For now, let’s get back to the ground. We still have to learn to crawl. After finishing up symmetry methods for first order ODEs in chapter 1, we shall proceed to extend (or these symmetry methods to higher order linear and nonlinear ODEs, and finally to linear and “prolong”) nonlinear PDEs. END Optional—6 Page Double Spaced PICTURE” Motivational digression Example with Discussion “BIG How is the infinitesimal generator Change of coordinates and the infinitesimal generator. affected by a change of coordinates? Suppose are new coordinates and let be an arbitrary smooth function. By the rule Without loss of generality, since is arbitrary, in the new coordinates
doctoral degree in physics, as it did with me. It’s physical underpinnings and mathematical grammar should be learned by the undergraduate between the sophomore and senior year, to be fully mastered during the first year and a half of graduate school where one should be learning the details of advanced mechanics, electrodynamics, quantum physics, thermodynamics and statistical physics, quantum electrodynamics, relativity, and possibly quantum fields with the power of a unified mathematical grammar and physical overview of where shit fits together and why it fits together. (Note—It doesn’t generally go backwards uniquely from postulating matrix representations back to (infinitesimal generator) realizations, back to differential equations, but this is nevertheless also a valid approach to investigate theoretical universes and extensions beyond the Standard Model.) There is, of course, more mathematics and physics beyond the unified outline expounded on in this work, but it ties back. For now, let’s get back to the ground. We still have to learn to crawl. After finishing up symmetry methods for first order ODEs in chapter 1, we shall proceed to extend (or “prolong”) these symmetry methods to higher order linear and nonlinear ODEs, and finally to linear and nonlinear PDEs. END Optional—6 Page Double Spaced “BIG PICTURE” Motivational digression Example with Discussion Change of coordinates and the infinitesimal generator. How is the infinitesimal generator affected by a change of coordinates? Suppose are new coordinates and let be an arbitrary smooth function. By the rule Without loss of generality, since is arbitrary, in the new coordinates