= Vector calculus
{wiki}
Vector calculus studies differentiation and integration of <scalar fields> and <vector fields>. Three operators answer different local questions: the <gradient> describes change of a scalar value, the <divergence> describes net outward flow, and the <curl> describes local circulation. Their associated integral theorems turn local derivatives into measurements along paths and across boundaries.
The difference between <divergence> and <curl> is especially visible in two smooth <vector fields> on three-dimensional <Euclidean space>:
$$
F(x,y,z)=(x,y,0),\qquad G(x,y,z)=(-y,x,0).
$$
At each point they have the same <Euclidean norm> $\sqrt{x^2+y^2}$, but their directions differ. For $F$, $\nabla\cdot F=2$ and $\nabla\times F=0$. For $G$, $\nabla\cdot G=0$ and $\nabla\times G=(0,0,2)$. One expands radially in each horizontal plane; the other describes rigid rotation.
\Image[/divergence-curl-flux.png]
{title=Equal-magnitude fields can have different divergence and curl: radial expansion gives outward flux, while rotation gives circulation}
Around the counterclockwise unit <circle> in the plane $z=0$, $F$ is normal to the boundary and has zero tangential component. Its outward planar <flux integral> is $2\pi$, while its <line integral> around the circle is zero. For $G$ the values are reversed: outward planar <flux> is zero, and circulation is $2\pi$. The <divergence> and <curl> predict exactly which boundary measurement survives.
Why is this useful? A <velocity field> can move material through a boundary or circulate it within a region. <Conservation of mass> connects its <divergence> with density changes through the <continuity equation>; its <curl> is <vorticity>. In <electromagnetism>, the <Maxwell equations> use the same operators to connect <electric fields>, <magnetic fields>, and their sources. The common mathematical language lets the same integral calculations work across different physical models. https://openstax.org/books/calculus-volume-3/pages/6-5-divergence-and-curl[OpenStax: Divergence, curl, and their interpretations]{ref}.
The geometric framework behind these operators is developed in <differential forms>. The <Generalized Stokes theorem> unifies the <fundamental theorem for line integrals>, <Green theorem>, <Stokes theorem>, and <divergence theorem>. In coordinates their formulas look different, but each relates accumulation over a region to a measurement on its boundary.
<Wilson's Vector Analysis>, published in 1901, was based on J. Willard Gibbs's lectures at Yale. Wilson's preface explains that examples from geometry, mechanics, and physics are interspersed with the mathematical development, so readers encounter applications as soon as they have the tools to understand them. That is a useful way to approach <vector calculus>: learn each operation through a measurement it makes possible. https://archive.org/details/vectoranalysiste00gibbiala[Wilson: Vector Analysis, 1901, title page and general preface]{ref}.
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