Created: June 07, 2024
Modified: June 08, 2024

vector divergence

This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.

The divergence of a vector-valued function $F$ on a vector field measures the extent to which a given point $\mathbf{x}$ is a source of the field.

It is defined as the flux along the boundary of an infinitesimal volume $V$ containing the point $\mathbf{x}$ . Formally, this is the surface integral of the quantity $F \cdot \hat{\mathbf{n}}$ where $\hat{\mathbf{n}}$ is the unit normal vector to the boundary:

\text{div}\; F = \lim_{|V|\to 0} \frac{1}{|V|} \oint_{S(V)} F(S) \cdot \hat{\mathbf{n}}(S) dS

Because everything gets locally linear when you get small enough, it turns out that the shape of the volume doesn't matter, so without loss of generality we can consider the axis-aligned cube of width $2\epsilon$ . This gives us an equation in Cartesian coordinates, first in one dimension

\text{div}\; F(x) = \lim_{\epsilon\to 0} \frac{1}{2\epsilon}\left[ F(\mathbf{x} - \epsilon) \cdot (-1) + F(\mathbf{x} + \epsilon) \cdot 1\right] = \frac{\partial F(x)}{\partial x}

and generalizing to multiple dimensions

\text{div}\; F(\mathbf{x}) = \sum_d \frac{\partial F_d(\mathbf{x})}{\partial x_d} = \nabla_x F_x + \nabla_y F_y + \ldots

which is sometimes written via abuse of notation as

\text{div} \; F(\mathbf{x}) = \nabla \cdot F.

vector divergence

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Laplacian

partial differential equation

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