Divergence is the Trace of the Jacobian

Saif Khattak · September 20, 2025

In seeking to better understand the basics of electromagnetism as described in Griffiths’ Introduction to Electrodynamics, I had some trouble understanding divergence. I had seen the diagrams—sources and sinks, very nice. I had heard many times: “divergence measures the rate at which a vector field appears to flow out of a certain region”. And I had even thought this meant I had understood divergence.

But the trouble to me was that these made no sense as descriptions of a generic function over a vector space. What does it mean for a function f: ℝⁿ → ℝᵐ to “flow out of a certain region”, and why should I believe that this is identical to the sum of its first order partial derivatives with respect to a given basis?

Well I won’t pretend to have a good explanation—I merely write this to point to the best one I’ve found: https://math.hawaii.edu/~lee/calculus/Curl.pdf and I quote:

The divergence of the vector field F, often denoted by ∇⋅F\nabla \cdot F, is the trace of the Jacobian matrix for FF, i.e. the sum of the diagonal elements of 𝒥.

Now this was intriguing, but still unintuitive. On further reading, I found a good explanation of the Jacobian in Tao’s Analysis II - the generalization of the single-variable derivative to a function f:Rn↦Rmf: ℝⁿ \mapsto ℝᵐ begins by asserting that ‘f being differentiable at a point x’ is equivalent to the assertion that it is ‘locally linearly approximable’. That is:

f(x⃗)≈f(x0⃗)+L(x⃗−x0⃗) f(\vec{x}) \approx f(\vec{x_0}) + L(\vec{x} - \vec{x_0})

for some linear map L. We know two things about L - it can be represented by a matrix (true of all linear maps), and it must map Rn↦Rm\mathbb{R}^n \mapsto \mathbb{R}^m (because x⃗−x0⃗\vec{x} - \vec{x_0} is in Rn\mathbb{R}^n). Section 6.2 of Tao spells it out, but this m×nm \times n matrix turns out to be exactly the Jacobian 𝒥.

We can now tie these together:

  • The Jacobian represents the local linear approximation of f(x⃗)f(\vec{x}).
  • For vector fields F:Rn↦RnF: \mathbb{R}^n \mapsto \mathbb{R}^n, the divergence is the trace of the Jacobian.
  • The trace of a square matrix is the sum of its eigenvalues.

Thus, divergence measures the infinitesimal rate of volume expansion - it tells us whether a small region is being stretched (positive divergence) or compressed (negative divergence) by the vector field.