Noether's Theorem: Symmetry and Conservation

A common calculus problem is to find an input that optimizes (maximizes or minimizes) a function. An extension of this problem is to find a function that optimizes an expression depending on the function. This paper studies how small (differentiable) variations of functions give us more information about expressions dependent on these functions. Specifically, Noether’s Theorem states that in a system of functions, each differential symmetry – or small variation where the system is invariant– constructs a conserved quantity. We will describe, interpret and prove Noether’s Theorem using techniques from linear algebra, differential geometry, and the calculus of variations. Furthermore, we will apply these techniques and Noether’s Theorem to physical examples such as the wave equation, the Schrödinger equation, and electromagnetism.