Mathematical Physics – notjustrocketscience

I have taught Mathematical Physics a few times. My favorite papers have been ‘Linear Vector Spaces‘ and ‘Complex Variables‘. My emphasis when teaching these has been to use these to illustrate how seemingly disparate concepts encountered in Physics can often be viewed as different manifestations of the same mathematical concepts. For instance, the theory of Special Functions can be viewed as eigenvalue problems of the Laplacian, or indeed orthonormal basis vectors in vector spaces of functions. Teaching Linear Vector Spaces has also allowed me to explore Spectral Theory relevant to Quantum Mechanics, Tangent Spaces in Differential Geometry and applications to Relativity (vectors in Spacetime). Also, a deeper understanding of Spectral Theory allows one to appreciate solutions to linear Differential Equations from the point of view of vector spaces.

My course on Complex Variables has been quite experimental, in which I have taken a geometrical approach, visualizing Complex Functions as maps involving translations, rotations and scaling transformations. I have discussed applications to Relativity (how does the Celestial Sphere appear to an Inertial Obeserver?), Integral Transforms (viewing Fourier and Laplace Transforms through Analytic Continuation).

Linear Vector Spaces
- Lectures on Linear Vector Spaces
- Application to Polynomial Approximation
Complex Analysis

Linear Vector Spaces

In this course, I apply the ideas developed to Partial Differential Equations (the Laplacian and its eigenvalues/eigenvectors related to Special Functions encountered so often in Physics), Differential Geometry (tangent vector spaces, the metric, etc), Relativity (four-vectors in spacetime and how their inner products can be related to measurable physical quantities (Doppler Shift, etc.)), Function Spaces in Quantum Mechanics (L2 spaces), etc. Some of these ideas , such as Function Spaces and Polynomials viewed as vectors belonging to sub-spaces of L2 spaces are applied to polynomial approximations to functions in a Computational Lab.

Lectures on Linear Vector Spaces

Linear_Master

Application to Polynomial Approximation

I love to apply the theory of Linear Vector Spaces to the problem of approximating functions with polynomials of a given degree. This is a problem similar to the ‘Method of Least Squares’, except applied to continuous set of values of a function. However, observing that functions act like ‘vectors’ over a domain with ‘inner product’ defined intuitively, and that polynomials of degree less than a certain value form a subspace of this vector space, we can use the method of ‘orthogonal projection’ developed in the lectures above to determine the polynomial which is ‘closest’ to a given function. Following is the technique I introduce to students

Orthogonal_Polynomials

Following is Python code to generate Legendre Polynomials (‘unit’ vectors in the polynomial subspace)

The following code fits an arbitrary function over an arbitrary interval to a polynomial using orthogonal projection