Gerald has taught engineering, math and science and has a doctorate in electrical engineering.
The curl is a vector operator in 3-dimensions. It measures the amount and direction of circulation in a vector field.
The steps to find the curl of a vector field:
You probably have seen the cross product of two vectors written as the determinant of a 3x3 matrix. We use this idea to write a general formula for the curl:
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The upside down triangle is called a ''nabla'' and the ''x'' suggests the cross product. We read the left-hand side as ''the curl of v''. On the right-hand side we see parameters labeled h, u and e. The v quantities we will get from the vector field.
How the vector field is written gives us the clues we need to identify the coordinate system:
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The parameters we need are :
For the Cartesian coordinate system,
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The parameter values for the cylindrical coordinate system:
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For the spherical coordinate system:
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For the Cartesian coordinate system, substituting:
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and then evaluating the determinant gives us the curl formula for the Cartesian coordinate system:
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For the cylindrical coordinate system
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Expanding the determinant:
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Simplifying, we get the curl formula in the cylindrical coordinate system:
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Finally, for the spherical coordinate system, substituting into the general curl equation:
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Expanding the determinant:
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Simplifying:
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Simplifying further gives the curl formula in the spherical coordinate system:
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The curl equation in each of our coordinate systems:
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Let's say we have a top view of a vector field with some easy-to-see circulation:
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The z axis points out of the plane. Using the right-hand rule convention, the clockwise circulation points into the x-y plane. Thus, we expect the direction of the curl to be in the negative z direction.
In Cartesian coordinates, this particular vector field v:
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In the Cartesian coordinate system, the curl formula is:
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Identify the vector components v1, v2 and v3:
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Evaluating all the required partial derivatives:
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Substituting into the curl formula:
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Simplifying:
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As expected, the circulation points along the -z direction and we see the magnitude is 2.
Find the curl of
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(This is the same v but expressed in cylindrical coordinates.)
Write the curl formula:
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Identifying the vector components:
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Evaluating all the required partial derivatives:
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Substituting into the curl formula:
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Simplifying:
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This is the same result as before because the z direction is the same in both the Cartesian and cylindrical coordinate systems. Usually, the choice of the coordinate system is selected to match the field. Otherwise, a consistent answer will be obtained but more work will be required. Let's explore this by calculating the curl of the same field but in the spherical coordinate system.
Find the curl of
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(Again, the same vector field but written in spherical coordinates.)
Write the curl formula:
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Identifying the vector components:
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Evaluating all the required partial derivatives:
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Substitute into the curl formula:
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Simplifying:
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This answer is more complicated because the symmetry of the vector field was not a good fit for spherical coordinates. Note the result depends only on the value of θ and is independent of r and φ. So, we will make sense of this result by checking some specific values of θ:
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The unit vector for the θ component points in the direction of increasing θ. At θ = 90o, this direction is in the negative z direction.
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This result also agrees with previous results: the curl points in the negative z direction and has a magnitude of 2.
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