《Alex_Alaniz_Lie(一步步学数学到博士)》.pdf

Forward. These are step-by-verifiable-step notes designed to take students with a year of calculus based physics who are about to enroll in ordinary differential equations all the way to doctoral foundations in either mathematics and physics without mystery. Abstract algebra, topology (local and global) folds into a useful, intuitive toolset for ordinary differential equations and partial differential equations, be they linear or nonlinear. The algebraist, the topologist, the theoretical physicist, the applied mathematician and experimental physicist are artificial distinctions at the core. There is unity. Mathematician, you will see step-by-verifiable-step algebra, topology (local and global) in a unified framework to treat differential equations, ordinary, partial, linear and nonlinear. You will then see why the physicists created a great font of differential equations, the calculus of variations. You will see why the physicists care about both discrete and continuous (topological) Lie groups and understand what quantum mechanics is as a mathematical system from its various historical classical physical roots: Lagrangian mechanics, Hamiltonian mechanics, Poisson brackets. You will have the tools to understand the Standard Model of physics and some of our main paths forward to grand unified theories and theories of everything. With these notes you should never again be able to practice abstraction for the sake of abstraction. Physicist, you will not be held hostage to verbiage and symbology. You will see that mathematics has deep, unavoidable limitations that underlie physics, itself suffering unavoidable limitations. You will see unity, e.g., summing angular momentum in terms of tensor products and directions sums, ladder operators, Young鈥檚 tableaux, root and weigh diagrams as different codifications of the same thing. Neither of you have to take your