Intersection of Eigenvectors and Multivariable Calculus This isn't really a problem but more of a reference/example question: do eigenvalues and eigenvectors ever show up in multivariable calculus? The two seem very unrelated to me. Specific examples would be appreciated.
Thanks in advance.
 A: Suppose $A$ is an $n\times n$ symmetric matrix and $x\in{\mathbb R}^n$.  Let $f(x)=x^TAx.$  When you use Lagrange multipliers to find the minimum and maximum of $f(x)$ subject to $\|x\|^2=1$, the eigenvalues show up as feasible Lagrange multipliers, and eigenvectors show up as vectors corresponding to the feasible Lagrange multipliers (in fact, you get the equation that is typically used to define eigenvalues and eigenvectors).  The maximum (minimum) of $f$ is the maximum (minimum) eigenvalue of $A$ and $f$ is maximized (minimized) by unit eigenvectors for this eigenvalue.
This is the only class of examples I'm aware of where the Lagrange multiplier itself has significance.
It's not the deepest example of what you're asking about, but there have been times where I've found the perspective of eigenvalues of $A$ as possible max/min values of $x^TAx$, given $\|x\|=1$ to be useful.
EDIT: I haven't studied these topics in depth enough to speak with any authority, but eigenvalues come up in relation to the shape operator, the Darboux frame, the Jacobian matrix, and the Hessian matrix, just to name a few things. You'd probably see the Jacobian and the Hessian in a typical multivariable calculus course.  You'd see the Darboux Frame and the shape operator in a differential geometry course.
