6.5 Divergence and curl  (Page 6/9)

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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 $\text{curl}\left(\text{∇}f\right)=0$ for any scalar function $f.$ In terms of our curl notation, $\nabla \times \nabla \left(f\right)=0.$ This equation makes sense because the cross product of a vector with itself is always the zero vector. Sometimes equation $\nabla \times \nabla \left(f\right)=0$ is simplified as $\nabla \times \nabla =0.$

Proof

Since $\text{curl}\text{F}=0,$ 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.

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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 $\text{curl}\text{F}=0.$

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.

Got questions? Get instant answers now!

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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 $\text{div}(\text{∇}f)=\nabla ·(\text{∇}f)=f_{xx}+f_{yy}.$ 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

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

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.

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.