Trace, Laplacian, the Heat equation, divergence theorem

The aim of this article is to help build an intuition for the trace of a matrix, "the sum of the elements on the diagonal" -- the basic idea is that the trace is an "average" of some sort, an average of the action of an operator or a quadratic form. We'll make this idea clearer with an example from classical physics: the heat equation.

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Consider an $n$-dimensional space with some temperature distribution $T(\vec{x},t)$. We wish to set up a differential equation for this function.

In the case that $n = 1$, this differential equation is exceedingly easy to write down, considering the difference $(T(x+dx)-T(x))-(T(x)-T(x-dx))$ as the double-derivative upon division by $dx^2$. More rigorously, what we're doing here is applying a localised version of the fundamental theorem of calculus. I.e. we're writing down:

$$\begin{align}
\lim_{\Delta x \to 0} \frac{1}{\Delta x}(T'(x + \Delta x) - T'(x)) &= \lim_{\Delta x \to 0} \frac{1}{{\Delta x}}\int_x^{\Delta x} {T''(x)dx}  \\
& = T''(x)
\end{align}
$$
More generally, we may consider the $n$-dimensional case.

Analogously to before, one may try to look at temperature flows in each direction -- here, we have an integral, done on the boundary of an infinitesimal region $V$ (this symbol will also represent the volume of the region):

$$ \frac{{\partial T}}{{\partial t}} = \lim_{V \to 0} \frac{\alpha }{V}\int_{\partial V} {\hat u\,dS \cdot \vec \nabla T} $$
At this point, one may apply the divergence theorem, converting this to:

$$\frac{{\partial T}}{{\partial t}} = \mathop {\lim }\limits_{V \to 0} \frac{\alpha }{V}\int\limits_V {\vec \nabla  \cdot \vec \nabla T\;dV}  = \alpha{\left| {\vec \nabla } \right|^2}T$$
In this sense, the divergence theorem is analogous to the fundamental theorem of calculus for manifolds with boundaries that are more than one-dimensional (see the bottom of the page for a link to a formalisation/an abstraction based on this analogy). But there are more ways to intuitively understand this. Note how the Laplacian is the trace of the Hessian matrix (note: we use $\vec{\nabla}^2$ to refer to the Hessian and $\left|\vec\nabla\right|^2$ to refer to the Laplacian):

$${\left| {\vec \nabla } \right|^2}T = {\mathop{\rm tr}} \left({\vec{\nabla} ^2}T\right)$$
The trace of a matrix is fundamentally linked to some notion of averaging -- the simplest interpretation of this is that it is the mean of the eigenvalues. But more relevant to our situation, it can be shown that the trace of a matrix is the expected value of the quadratic form defined by the matrix on the unit sphere -- or on a general sphere $S$:

$${\mathop{\rm tr}} A = \frac{1}{S}\int_S {\frac{{\Delta {x^T}A\,\Delta x}}{{\Delta {x^T}\Delta x}}\,dS} $$
One may check that taking the limit as $\Delta x \to 0$, substituting $\nabla^2$ for the operator and writing ${\overrightarrow \nabla ^2}f\,d\vec x = \overrightarrow \nabla  f$, one gets the original "average of directional derivatives" expression.

Can you interpret the other coefficients of the characteristic polynomial in terms of statistical ideas?

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Further reading:

• Using the "infinitesimal region" idea to define divergence, curl and Laplacian rigorously: Khan Academy
• An abstraction based on the "analogy" between FTC, Divergence Theorem, Navier-Stokes Theorem, etc. Stokes' theorem (Wikipedia)