Term

Definition

Conservative Vector Field F

Irrotational Vector Field F

Path-Independent Property of F

Closed-Loop Property of F

Exact (or Total) Differential

Solenoidal Vector Field F

Scalar Potential for F

Vector Potential for F

Harmonic Function

A scalar function $u$ whose Laplacian vanishes everywhere is said to be "harmonic." Thus, $u$ is harmonic if ${\nabla }^{2}u\= \nabla ·\left(\nabla u\right)\=0$ everywhere.

Table 9.7.1   Glossary of relevant terms

Identity

Mnemonic

$\nabla ·\left(\nabla \times \mathbf{F}\right)\equiv 0$

Curls don't spread.

$\nabla \times \left(\nabla u\right)\equiv \mathbf{0}$

Gradients don't twist.

Table 9.7.2   Two  differential identities

Characterizations of Conservative Vector Field F

1.

$\mathbf{F}\= \nabla u$ 

2.

F is irrotational

3.

The expression $\mathbf{F}·\mathbf{dr}$ is exact

4.

F has the Path-Independence property

${\int }_C\mathbf{F}·\mathbf{dr}\={\int }_P^Q\mathrm{du}\=u\left(Q\right)-u\left(P\right)$ 

and the integral is independent of the path $C$. The converse is also true, so that path independence implies the existence of a scalar potential $u$, making F conservative.

5.

F has the Closed-Loop property

Table 9.7.3   Characterizations of "Conservative"

Recipe 1

$u\left(x\,y\,z\right)\={\int }_a^{x}f\left(t\,b\,c\right) \mathrm{dt}\+{\int }_b^{y}g\left(x\,t\,c\right) \mathrm{dt}\+{\int }_c^{z}h\left(x\,y\,t\right) \mathrm{dt}$ 

Recipe 2

$\mathbf{A}\=\left({\int }_a^{x}h\left(t\,y\,z\right) \mathrm{dt}-{\int }_c^{z}f\left(a\,y\,t\right) \mathrm{dt}\right) \mathbf{j}-\left({\int }_a^{x}g\left(t\,y\,z\right) \mathrm{dt}\right) \mathbf{k}$

Table 9.7.4   Recipes for obtaining scalar and vector potentials

Example 9.7.1

Show that the line integral of $\mathbf{F}\=\left(2 x y-y^3\right) \mathbf{i}\+\left(x^2-3 x y^2\right) \mathbf{j}$ over any circle in the plane is zero.

Example 9.7.2

Prove that $\mathbf{F}\=\left(2 x y-y^3\right) \mathbf{i}\+\left(x^2-3 x y^2\right) \mathbf{j}$ is conservative.

Example 9.7.3

Find a scalar potential for $\mathbf{F}\=\left(2 x y-y^3\right) \mathbf{i}\+\left(x^2-3 x y^2\right) \mathbf{j}$.

Example 9.7.4

If $u\left(x\,y\right)$ is a scalar potential for $\mathbf{F}\=\left(2 x y-y^3\right) \mathbf{i}\+\left(x^2-3 x y^2\right) \mathbf{j}$, show that ${\int }_C\mathbf{F}·\mathbf{dr}\=u\left(Q\right)-u\left(P\right)$, where $C$ is that part of the parabola $y\=x^2$ between P and Q, the points $\left(-2\,4\right)$, and $\left(2\,4\right)$, respectively.

Example 9.7.5

For $\mathbf{F}\=\left(2xz\+y^2\right) \mathbf{i}\+\left(2 x y-z^3\right) \mathbf{j}\+\left(x^2-3y{z}^2\right) \mathbf{k}$; $C_1$, the line from the origin to the point $\left(1\,1\,1\right)$; and $C_2$, the polygonal path from the origin to $\left(1\,0\,0\right)$ to $\left(1\,1\,0\right)$, to $\left(1\,1\,1\right)$, show that the line integral of F along $C_1$ and $C_2$ have the same value.

Example 9.7.6

For $\mathbf{F}\=\left(2xz\+y^2\right) \mathbf{i}\+\left(2 x y-z^3\right) \mathbf{j}\+\left(x^2-3y{z}^2\right) \mathbf{k}$, show that the line integral along any member of the family of curves $x\=t\,y\=t^k\,z\=t^{2 k}\,t\in \left[0\,1\right]$, $k\ge 1$, has the value 1.

Example 9.7.7

Prove that $\mathbf{F}\=\left(2xz\+y^2\right) \mathbf{i}\+\left(2 x y-z^3\right) \mathbf{j}\+\left(x^2-3y{z}^2\right) \mathbf{k}$ is conservative by showing it is curl free.

Example 9.7.8

Find a scalar potential for $\mathbf{F}\=\left(2xz\+y^2\right) \mathbf{i}\+\left(2 x y-z^3\right) \mathbf{j}\+\left(x^2-3y{z}^2\right) \mathbf{k}$.

Example 9.7.9

Show that $\mathbf{F}\=\left(2xz\+y^2\right) \mathbf{i}\+\left(2 x y-z^3\right) \mathbf{j}\+\left(x^2-3y{z}^2\right) \mathbf{k}$ is not solenoidal.

Example 9.7.10

If $u\left(x\,y\,z\right)$ is a scalar potential for

$\mathbf{F}\=\left(2xz\+y^2\right) \mathbf{i}\+\left(2 x y-z^3\right) \mathbf{j}\+\left(x^2-3y{z}^2\right) \mathbf{k}$ 

show that ${\int }_C\mathbf{F}·\mathbf{dr}\=u\left(Q\right)-u\left(P\right)$, where $C$ is given parametrically by the equations $x\=1\+2 t\,y\=2 t^2-t\+3\,z\=2\+t^3$, and P and Q are its endpoints when $t\=0\,1$, respectively.

Example 9.7.11

Show that $\mathbf{F}\=x z \mathbf{i}\+\left(x^2-y z\right) \mathbf{j}\+y^2 \mathbf{k}$ is solenoidal but not conservative, and find a vector potential both with Maple's VectorPotential command and by Recipe 2 in Table 9.7.4. If these differ, show that the difference is a gradient and find a scalar potential for this gradient.

Example 9.7.12

Show that $\mathbf{F}\=3\left(x^2-y^2\right) \mathbf{i}-6 x y \mathbf{j}$ is both solenoidal and conservative. Find a scalar potential and a vector potential. Show that the scalar potential is harmonic.