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curl F = ( R y Q z ) i + ( P z R x ) j + ( Q x P y ) k = 0.

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 curl ( f ) = 0 for any scalar function f . In terms of our curl notation, × ( f ) = 0 . This equation makes sense because the cross product of a vector with itself is always the zero vector. Sometimes equation × ( f ) = 0 is simplified as × = 0 .

Curl test for a conservative field

Let F = P , Q , R be a vector field in space on a simply connected domain. If curl F = 0 , then F is conservative.


Since curl 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 curl F = 0 .

Testing whether a vector field is conservative

Use the curl to determine whether F ( x , y , z ) = y z , x z , x y is conservative.

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

A diagram showing the curl of a vector field in two dimensions. The curl is zero. The arrows seem to be pointing up and over into the yz plane.
The curl of vector field F ( x , y , z ) = y z , x z , x y is zero.

The curl of F is

( y x y z x z ) i + ( y y z z x y ) j + ( y x z z y z ) k = ( x x ) i + ( y y ) j + ( z z ) k = 0 .

Thus, F is conservative.

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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 div ( f ) = · ( f ) = f x x + f y y . We abbreviate this “double dot product” as 2 . This operator is called the Laplace operator , and in this notation Laplace’s equation becomes 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

div ( f ) = · ( f ) = f x x + f y y + f z z .

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

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 ( x , y ) = x 2 + x y to be the potential function of an electrostatic field that is located in a region of 2 free of static charge?

If f were such a potential function, then f would be harmonic. Note that f x x = 2 and f y y = 0 , and so f x x + f y y 0 . Therefore, f is not harmonic and f cannot represent an electrostatic potential.

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Is it possible for function f ( x , y ) = x 2 y 2 + x to be the potential function of an electrostatic field located in a region of 2 free of static charge?


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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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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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