Scalar, Vector, Flux, and Complex Line Integrals

All flashcards

What does a scalar line integral accumulate?

Values of a scalar field along a curve, weighted by arc length: $\int_C f\,ds$.

How is a vector line integral computed from a parametrized curve?

Use $\int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt$; the dot product selects the field’s tangential component.

What is a potential function for a conservative vector field?

A scalar function $G$ whose gradient equals the field: $\mathbf{F}=\nabla G$.

What perpendicular vector gives planar flux in the clockwise convention?

For velocity $(x',y')$, use $(y',-x')$, giving flux integrand $Py'-Qx'$ for $\mathbf{F}=(P,Q)$.

How is a complex line integral evaluated along $\gamma(t)$?

Use $\int_a^b f(\gamma(t))\gamma'(t)\,dt$.

Quiz questions

1. For a force field $\mathbf{F}$ acting along a directed path $C$, which integral gives the work done?Answer: $\int_C \mathbf{F}\cdot d\mathbf{r}$
2. What happens to a vector line integral when the direction of traversal is reversed?Answer: Its sign is reversed.
3. If $\mathbf{F}=\nabla G$, how can the line integral from point $A$ to point $B$ be evaluated?Answer: $G(B)-G(A)$
4. For a complex function $f$ along a parametrized curve $\gamma(t)$, which expression gives $\int_L f(z)\,dz$?Answer: $\int_a^b f(\gamma(t))\gamma'(t)\,dt$
5. Under the Cauchy integral theorem, when is the integral of an analytic function around a suitable closed curve zero?Answer: When the function is analytic throughout the enclosed region.