TITLE: What is the intuition behind the trace of an endomorphism?
QUESTION [11 upvotes]: Looking back on my math education, I noticed that even though the trace of an endomorphism came up a lot, I'd be hard pressed to give a good description of what the trace really means. I'm looking for some good intuition about its meaning.
To elaborate on what I'm looking for: if I forgot the rigorous definition of the determinant, I could rebuild it from scratch without reference to a basis because I know that it's supposed to measure the change in volume and orientation of a parallelepiped under a linear transformation. For the same reason, I can quickly tell that it is independent of basis, multiplicative, and determines wether the endomorphism is injective or not, all without doing any calculations. I want something similar for the trace. It doesn't need to be geometric, but I want to know what the trace tells us about how the endomorphism acts.

REPLY [2 votes]: Let $f:U\subset \mathbb R^n\to\mathbb R^n$  be a $C^1$ vector field, then the trace of differential of $f$ (or trace of jacobian matrix), is equal to, the divergence of $f$. Explicitly, $$\operatorname{trace}\big(df|_p\big)=\operatorname{trace}\left(\array{\frac{\partial f^1}{\partial x^1} & \cdots & \frac{\partial f^1}{\partial x^n}\\ \vdots & \ddots & \vdots \\ \frac{\partial f^n}{\partial x^1}& \cdots & \frac{\partial f^n}{\partial x^n}}\right)|_p=\sum_{i=1}^n \frac{\partial f^i}{\partial x^i}|_p=\operatorname{div} f|_p$$
And divergence is coordinate free in the sense of Euclid inner space $\mathbb R^n$ or $\operatorname{SO}(n)$, because
$$\operatorname{div} f|_p=\lim_{r\to 0}\frac{1}{\operatorname{Vol}(U_r)}\int_{\partial U_r} \langle f,\mathbf{n} \rangle(x)\, dx$$
where (1) $U_r$ can be a $r$-radius ball or rectangle that center at $p$, (2) $\mathbf{n}$ is outer unit normal vector of surface $\partial U_r$ at $x$, (3) $\langle f,\mathbf{n} \rangle(x)$ is the inner product at $x\in \partial U_r$, (4) surface integral.