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Chapter 9: Vector Calculus
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Section 9.7: Conservative and Solenoidal Fields
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Essentials


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Table 9.7.1 defines a number of relevant terms.
Term

Definition

Conservative Vector Field F

•

A conservative field F is a gradient of some scalar, do that $\mathbf{F}\=\nabla u$.

•

In physics, because of the connection of the scalar $u$ to potential energy, the conservative field is typically taken as $\mathbf{F}\=\nabla u$.

•

The scalar $u$ is called the "scalar potential" for F.


Irrotational Vector Field F

•

An irrotational field is one for which the curl vanishes everywhere.
Hence, F is irrotational if $\nabla \times \mathbf{F}\equiv \mathbf{0}$.

•

Irrotational fields are also said to be "curl free."


PathIndependent Property of F

•

A vector field F for which the line integral between any two points P and Q has the same value over any curve connecting P and Q is said to have the "PathIndependent" property.


ClosedLoop Property of F

•

A vector field F for which the line integral around any closed path is zero is said to have the "ClosedLoop" property.

•

The ClosedLoop property and the PathIndependent property are equivalent. If F has one such property, it automatically has the other.


Exact (or Total) Differential

•

As per Section 4.11, if a scalar $f\left(x\,y\,z\right)$ is differentiable, it has a differential that can be written as $\mathrm{df}\={f}_{x}\mathrm{dx}plus;{f}_{y}\mathrm{dy}plus;{f}_{z}\mathrm{dz}$.

•

Given the arbitrary collection of symbols ${u}_{x}\mathrm{dx}plus;{u}_{y}\mathrm{dy}plus;{u}_{z}\mathrm{dz}$, the claim that this is an exact or total differential is the claim that a differentiable function $u\left(x\,y\,z\right)$ exists, and the given expression is its differential.


Solenoidal Vector Field F

•

A vector field F whose divergence vanishes everywhere ($\nabla \xb7\mathbf{F}\equiv 0$) is called a solenoidal field.

•

In other words a solenoidal field is "divergence free."


Scalar Potential for F

•

If F is the gradient of some scalar function, that scalar function is called the "scalar potential" for F.


Vector Potential for F

•

If F is the curl of a vector field A, then A is said to be a "vector potential" for F.


Harmonic Function

A scalar function $u$ whose Laplacian vanishes everywhere is said to be "harmonic." Thus, $u$ is harmonic if ${\nabla}^{2}u\=\nabla \xb7\left(\nabla u\right)equals;0$ everywhere.

Table 9.7.1 Glossary of relevant terms



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Recall Identities 1 and 2 in Table 9.4.1. For the reader's convenience, these two differential identities are listed again in Table 9.7.2, along with the mnemonics that appear in the paragraphs below the original table.
Identity

Mnemonic

$\nabla \xb7\left(\nabla \times \mathbf{F}\right)\equiv 0$

Curls don't spread.

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

Gradients don't twist.

Table 9.7.2 Two differential identities



Define a "curl" for planar vector fields $\mathbf{F}\left(x\,y\right)\=f\left(x\,y\right)\mathbf{i}plus;g\left(xcomma;y\right)\mathbf{j}$ by adding a third component $0\mathbf{k}$. Vanishing of the curl is then equivalent to the identity ${f}_{y}\={g}_{x}$. It is then convenient to use the terms "curl free" and "irrotational" for any Cartesian field in two or three dimensions.
Table 9.7.3 lists characterizations of conservative vector fields. In other words, a vector field F with any one of the characteristics listed in Table 9.7.3 is necessarily conservative, and a conservative field F has every one of the properties in the table. Because it appears so often in the literature, the symbol dr is taken as the differential of the position vector $\mathbf{r}\=x\mathbf{i}plus;y\mathbf{j}plus;z\mathbf{k}$. Since $\mathbf{dr}\=\mathrm{dx}\mathbf{i}plus;\mathrm{dy}\mathbf{j}plus;\mathrm{dz}\mathbf{k}$ is itself a vector, this ebook will use the notation dr, whereas some texts might write $d\mathbf{r}$.
Characterizations of Conservative Vector Field F

1.

$\mathbf{F}\=\nabla u$


2.

F is irrotational

•

Differential Identity 2 (gradients don't twist) shows that any vector field that is curl free (i.e., irrotational) must necessarily be a gradient. Hence, that field is conservative.


3.

The expression $\mathbf{F}\xb7\mathbf{dr}$ is exact

•

If $\mathbf{F}\xb7\mathbf{dr}\=a\mathrm{dx}plus;b\mathrm{dy}plus;c\mathrm{dz}$ is exact, then there exists a differentiable function $u$ whose first partial derivatives are $a\,b\,c$, respectively, so $\mathbf{F}\={u}_{x}\mathbf{i}plus;{u}_{y}\mathbf{j}plus;{u}_{z}\mathbf{k}equals;\nabla u$, and F is necessarily conservative.


4.

F has the PathIndependence property

•

If F is conservative, then from Characterization 3,

${\int}_{C}\mathbf{F}\xb7\mathbf{dr}\={\int}_{P}^{Q}\mathrm{du}\=u\left(Q\right)u\left(P\right)$
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 ClosedLoop property

•

On any closed path $C$, pick two points P and Q. This separates the loop into two paths, ${c}_{1}$ clockwise from P to Q and ${c}_{2}$, counterclockwise from P to Q.

•

The integral around the loop is the sum of the integral over ${c}_{1}$ plus the negative of the integral over ${c}_{2}$. But, by the ClosedLoop property, this sum is zero, so the integrals over ${c}_{1}$ and ${c}_{2}$ must be equal, and F has the PathIndependence property.


Table 9.7.3 Characterizations of "Conservative"



Characterization 2 in Table 9.7.3 provides a computational way to test if F is conservative. Any vector field whose curl vanishes identically is necessarily conservative.
Characterization 3 in Table 9.7.3 is the basis for what some texts call the "Fundamental Theorem for Line Integrals." A loose statement of such a theorem might be "the line integral of the tangential component of the gradient $\nabla u$ equals the difference in the endpoint values of $u$."
Characterization 5 in Table 9.7.3 shows that if F has the ClosedLoop property, then it has the PathIndependence property. If F has the PathIndependence property, then it has the ClosedLoop property by similar manipulations. For example, let ${c}_{1}$ and ${c}_{2}$ be two distinct paths from P to Q. If the integral from P to Q over each path has the same value, then since the integral from Q to P over, say, ${c}_{2}$ is the negative of the integral from P to Q, integrating around the closed loop from P to Q and back to P by traversing ${c}_{1}$ and then ${c}_{2}$ must necessarily have the value zero.${}$
Differential Identity 1 in Table 9.7.2 provides a characterization of solenoidal (divergencefree) fields. Since the divergence of any curl is zero, then any such curl is necessarily a solenoidal field.
•

Certain vector fields are both conservative and solenoidal. These are fields whose scalar potential is harmonic.

•

Figure 9.7.1 is a Venn diagram showing the relationship between conservative and solenoidal fields.

•

Those fields that are both conservative (C) and solenoidal (S) have a scalar potential that is harmonic (H).


>

use plots, plottools in
module()
local p1,p2,p3,p4,p5,p6;
p1 := circle([0,0],1):
p2 := ellipse([.9,.5],.8,.6):
p3 := rectangle([2,1.5],[2,1.5], style=line):
p4 := textplot({[.5,.5,"C"], [.45,.55,"H"], [1.2,.5,"S"],[1.5,.75,"N"]}, font=[TIMES,BOLD,14]):
p5 := textplot({[.4,0,"Conservative"],[1,1.1,"Not Conservative"],[.58,.25,`Harmonic`],[1,.9,"Solenoidal"]}, font=[TIMES,ROMAN,10]):
p6 := display([p1,p2,p3,p4,p5], scaling=constrained, axes=none):
print(p6);
end module:
end use:


Figure 9.7.1 Conservative vs. Solenoidal






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Table 9.7.4 provides Recipe 1 for finding a scalar potential, and Recipe 2 for finding a vector potential for the vector field $\mathbf{F}\=f\left(x\,y\,z\right)\mathbf{i}plus;g\left(xcomma;ycomma;z\right)\mathbf{j}plus;h\left(xcomma;ycomma;z\right)\mathbf{k}$. For Recipe 1 to apply, the curl of F must vanish. For Recipe 2 to apply, the divergence of F must vanish.
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Recipe 1

$u\left(x\,y\,z\right)\={\int}_{a}^{x}f\left(t\,b\,c\right)\mathrm{dt}plus;{\int}_{b}^{y}g\left(xcomma;tcomma;c\right)\mathrm{dt}plus;{\int}_{c}^{z}h\left(xcomma;ycomma;t\right)\mathrm{dt}$

Recipe 2

$\mathbf{A}\=\left({\int}_{a}^{x}h\left(t\,y\,z\right)\mathrm{dt}{\int}_{c}^{z}f\left(acomma;ycomma;t\right)\mathrm{dt}\right)\mathbf{j}\left({\int}_{a}^{x}g\left(tcomma;ycomma;z\right)\mathrm{dt}\right)\mathbf{k}$

Table 9.7.4 Recipes for obtaining scalar and vector potentials



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If F has a scalar potential $u\left(x\,y\,z\right)$, it is unique up to an additive constant. Thus, if there is another scalar potential $v\left(x\,y\,z\right)$, then $uv$ is necessarily a constant.
If F has a vector potential A, it is unique up to the addition of a gradient. Thus, if there is another vector potential B, then the difference $\mathbf{C}\=\mathbf{A}\mathbf{B}$ is itself a gradient vector, so that $\nabla \times \mathbf{C}\=\mathbf{0}$, that is, the curl of C must vanish.
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Examples


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Example 9.7.1

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

Example 9.7.2

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

Example 9.7.3

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

Example 9.7.4

If $u\left(x\,y\right)$ is a scalar potential for $\mathbf{F}\=\left(2xy{y}^{3}\right)\mathbf{i}plus;\left({x}^{2}3x{y}^{2}\right)\mathbf{j}$, show that ${\int}_{C}\mathbf{F}\xb7\mathbf{dr}\=u\left(Q\right)u\left(P\right)$, 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 $\mathbf{F}\=\left(2xz\+{y}^{2}\right)\mathbf{i}plus;\left(2xy{z}^{3}\right)\mathbf{j}plus;\left({x}^{2}3y{z}^{2}\right)\mathbf{k}$; ${C}_{1}$, 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 $\mathbf{F}\=\left(2xz\+{y}^{2}\right)\mathbf{i}plus;\left(2xy{z}^{3}\right)\mathbf{j}plus;\left({x}^{2}3y{z}^{2}\right)\mathbf{k}$, show that the line integral along any member of the family of curves $x\=t\,y\={t}^{k}\,z\={t}^{2k}comma;t\in \left[0comma;1\right]$, $k\ge 1$, has the value 1.

Example 9.7.7

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

Example 9.7.8

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

Example 9.7.9

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

Example 9.7.10

If $u\left(x\,y\,z\right)$ is a scalar potential for
$\mathbf{F}\=\left(2xz\+{y}^{2}\right)\mathbf{i}plus;\left(2xy{z}^{3}\right)\mathbf{j}plus;\left({x}^{2}3y{z}^{2}\right)\mathbf{k}$
show that ${\int}_{C}\mathbf{F}\xb7\mathbf{dr}\=u\left(Q\right)u\left(P\right)$, where $C$ is given parametrically by the equations $x\=1\+2tcomma;yequals;2{t}^{2}tplus;3comma;zequals;2plus;{t}^{3}$, and P and Q are its endpoints when $t\=0\,1$, respectively.

Example 9.7.11

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



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