11 Theorems of Vector Calculus Free Online FlashCards

What does circulation measure?

Circulation is the line integral $\oint_C \mathbf F\cdot d\mathbf r$, measuring the field’s tangential accumulation around an oriented curve.

What does flux measure?

Flux is $\iint_S \mathbf F\cdot\mathbf n\,dS$, measuring how much of a vector field crosses an oriented surface.

How is divergence computed?

For $\mathbf F=\langle P,Q,R\rangle$, $\nabla\cdot\mathbf F=P_x+Q_y+R_z$.

What is curl’s local meaning?

The curl component $(\nabla\times\mathbf F)\cdot\mathbf n$ is the limiting circulation per unit area for small surfaces normal to $\mathbf n$.

State Stokes’ theorem.

Stokes’ theorem: $\oint_C\mathbf F\cdot d\mathbf r=\iint_S(\nabla\times\mathbf F)\cdot d\mathbf S$, where $C=\partial S$ has compatible orientation.

How is Stokes orientation determined?

The right-hand rule sets the positive orientation: curl your right-hand fingers in the traversal direction; your thumb points along the surface normal.

When may a Stokes surface change?

A Stokes surface can be replaced by any easier spanning surface with the same oriented boundary, provided the hypotheses are satisfied.

State the divergence theorem.

The divergence theorem states $\iint_S\mathbf F\cdot d\mathbf S=\iiint_E\nabla\cdot\mathbf F\,dV$, for a closed surface $S=\partial E$ oriented outward.

Can divergence theorem use an open surface?

No. The divergence theorem requires a closed surface. An open surface must be completed, and the added boundary’s flux must be handled separately.

Find the outward flux of $\langle x,y,z\rangle$ on a sphere.

For $\mathbf F=\langle x,y,z\rangle$ on a sphere of radius $R$, $\nabla\cdot\mathbf F=3$, so the outward flux is $3\cdot\frac{4}{3}\pi R^3=4\pi R^3$.

What does a gradient field imply about circulation?

If $\mathbf F=\nabla f$ and the relevant derivatives are continuous, then $\nabla\times\mathbf F=\mathbf0$, so circulation around every suitable closed curve is zero.

What is the local conservation law?

The local conservation law is $\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf J=0$, expressing that density changes through net flux.