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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.

Proof

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?

Yes

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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.

Questions & Answers

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20/(×-6^2)
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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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