# 6.5 Divergence and curl  (Page 6/9)

 Page 6 / 9
$\begin{array}{cc}\hfill \text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}& =\left({R}_{y}-{Q}_{z}\right)\text{i}+\left({P}_{z}-{R}_{x}\right)\text{j}+\left({Q}_{x}-{P}_{y}\right)\text{k}\hfill \\ & =0.\hfill \end{array}$

The same theorem is true for vector fields in a plane.

Since a conservative vector field is the gradient of a scalar function, the previous theorem says that $\text{curl}\phantom{\rule{0.2em}{0ex}}\left(\text{∇}f\right)=0$ for any scalar function $f.$ In terms of our curl notation, $\nabla \phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\nabla \left(f\right)=0.$ This equation makes sense because the cross product of a vector with itself is always the zero vector. Sometimes equation $\nabla \phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\nabla \left(f\right)=0$ is simplified as $\nabla \phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\nabla =0.$

## Curl test for a conservative field

Let $\text{F}=⟨P,Q,R⟩$ be a vector field in space on a simply connected domain. If $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=0,$ then F is conservative.

## Proof

Since $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=0,$ we have that ${R}_{y}={Q}_{z},{P}_{z}={R}_{x},$ and ${Q}_{x}={P}_{y}.$ Therefore, F satisfies the cross-partials property on a simply connected domain, and [link] implies that F is conservative.

The same theorem is also true in a plane. Therefore, if F is a vector field in a plane or in space and the domain is simply connected, then F is conservative if and only if $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=0.$

## Testing whether a vector field is conservative

Use the curl to determine whether $\text{F}\left(x,y,z\right)=⟨yz,xz,xy⟩$ is conservative.

Note that the domain of F is all of ${ℝ}^{3},$ which is simply connected ( [link] ). Therefore, we can test whether F is conservative by calculating its curl.

The curl of F is

$\left(\frac{\partial }{\partial y}xy-\frac{\partial }{\partial z}xz\right)\text{i}+\left(\frac{\partial }{\partial y}yz-\frac{\partial }{\partial z}xy\right)\text{j}+\left(\frac{\partial }{\partial y}xz-\frac{\partial }{\partial z}yz\right)\text{k}=\left(x-x\right)\text{i}+\left(y-y\right)\text{j}+\left(z-z\right)\text{k}=0.$

Thus, F is conservative.

We have seen that the curl of a gradient is zero. What is the divergence of a gradient? If $f$ is a function of two variables, then $\text{div}\left(\text{∇}f\right)=\nabla ·\left(\text{∇}f\right)={f}_{xx}+{f}_{yy}.$ We abbreviate this “double dot product” as ${\nabla }^{2}.$ This operator is called the Laplace operator , and in this notation Laplace’s equation becomes ${\nabla }^{2}f=0.$ Therefore, a harmonic function is a function that becomes zero after taking the divergence of a gradient.

Similarly, if $f$ is a function of three variables then

$\text{div}\left(\text{∇}f\right)=\nabla ·\left(\text{∇}f\right)={f}_{xx}+{f}_{yy}+{f}_{zz}.$

Using this notation we get Laplace’s equation for harmonic functions of three variables:

${\nabla }^{2}f=0.$

Harmonic functions arise in many applications. For example, the potential function of an electrostatic field in a region of space that has no static charge is harmonic.

## Finding a potential function

Is it possible for $f\left(x,y\right)={x}^{2}+x-y$ to be the potential function of an electrostatic field that is located in a region of ${ℝ}^{2}$ free of static charge?

If $f$ were such a potential function, then $f$ would be harmonic. Note that ${f}_{xx}=2$ and ${f}_{yy}=0,$ and so ${f}_{xx}+{f}_{yy}\ne 0.$ Therefore, $f$ is not harmonic and $f$ cannot represent an electrostatic potential.

Is it possible for function $f\left(x,y\right)={x}^{2}-{y}^{2}+x$ to be the potential function of an electrostatic field located in a region of ${ℝ}^{2}$ free of static charge?

Yes

## Key concepts

• The divergence of a vector field is a scalar function. Divergence measures the “outflowing-ness” of a vector field. If v is the velocity field of a fluid, then the divergence of v at a point is the outflow of the fluid less the inflow at the point.
• The curl of a vector field is a vector field. The curl of a vector field at point P measures the tendency of particles at P to rotate about the axis that points in the direction of the curl at P .
• A vector field with a simply connected domain is conservative if and only if its curl is zero.

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Salomon
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Salomon
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Salomon
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