Vector Analysis

Syllabus: Vectors in plane and space, vector operations, gradient, divergence and curl, Gauss's, Green's and Stokes' theorems.

7 hand-picked visual resources on this topic, 1 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Gradient, divergence, curl

Gradient acts on a scalar and gives a vector; divergence acts on a vector and gives a scalar; curl gives a vector. Two identities are worth knowing cold because they kill options instantly.

$\nabla f=\left(f_x,f_y,f_z\right)$
points along steepest increase
$\nabla\cdot\vec F=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$
$\nabla\times(\nabla f)=0,\qquad \nabla\cdot(\nabla\times\vec F)=0$

See it

Directional derivative

The rate of increase along a direction is the gradient dotted with a unit vector in that direction.

$D_{\hat u}f=\nabla f\cdot\hat u,\qquad \hat u=\frac{\vec v}{\lVert\vec v\rVert}$

Gauss, Stokes and Green

Each converts an integral over a boundary into one over the interior. Choosing the easier side is usually the whole question.

$\oiint_S\vec F\cdot d\vec S=\iiint_V(\nabla\cdot\vec F)\,dV$

Conservative fields

If $\nabla\times\vec F=0$ the field is a gradient, the line integral depends only on the endpoints, and any closed-loop integral is zero — whatever the contour looks like.

$\nabla\times\vec F=0\Rightarrow\oint_C\vec F\cdot d\vec l=0$

← Differential Equations Complex Analysis →

Loading revision…