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Finding the Curl of a Vector Field: Steps & How-to

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson you will find the curl of a vector field in three different coordinate systems. A method for generating the curl formula in each of these coordinate systems is also presented.

Find the Curl of a Vector Field

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:

Step 1: Use the general expression for the curl.

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:


general_way_to_write_curl_formula


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.

Step 2: Identify the coordinate system.

How the vector field is written gives us the clues we need to identify the coordinate system:

  • The basis vectors i, j and k along with the coordinates x, y and z identify the Cartesian coordinate system.
  • The basis vectors eρ, eφ and ez along with the coordinates ρ, φ and z indicate the cylindrical coordinate system.


Relationship between Cartesian and cylindrical coordinate systems
relationship_between_Cartesian_and_cylindrical


  • In the spherical coordinate system, r is the distance from the origin directly to the point P. The two other coordinates are angles, θ and φ.


Relationship between Cartesian and spherical
Cartesian_and_spherical


Step 3: Look up the parameters for the identified coordinate system.

The parameters we need are :

  • the coordinates, u
  • the basis vectors, e
  • the appropriate multipliers, h

For the Cartesian coordinate system,


parameters_Cartesian


The parameter values for the cylindrical coordinate system:


parameters_cylindrical


For the spherical coordinate system:


parameters_spherical


Step 4: Substitute the parameters into the general equation, evaluate the determinant and simplify.

For the Cartesian coordinate system, substituting:


substituting_parameters


and then evaluating the determinant gives us the curl formula for the Cartesian coordinate system:


the_curl_formula_in_Cartesian_coordinates


For the cylindrical coordinate system


substituting_cylindrical_parameters


Expanding the determinant:


expanded_determinant_curl_cylindrical_coordinates


Simplifying, we get the curl formula in the cylindrical coordinate system:


the_curl_formula_in_cylindrical_coordinates


Finally, for the spherical coordinate system, substituting into the general curl equation:


substituting_spherical_parameters


Expanding the determinant:


expanded_determinant_curl_spherical_coordinates


Simplifying:


simplifying_determinant_curl_spherical_coordinates


Simplifying further gives the curl formula in the spherical coordinate system:


curl_spherical_coordinates


The Final Results

The curl equation in each of our coordinate systems:

Cartesian Coordinate System


the_curl_formula_in_Cartesian_coordinates


Cylindrical Coordinate System


the_curl_formula_in_cylindrical_coordinates


Spherical Coordinate System


curl_spherical_coordinates


An Application

Let's say we have a top view of a vector field with some easy-to-see circulation:


Clockwise circulation in a vector field
clockwise_circulation


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.

The Curl in Cartesian Coordinates

In Cartesian coordinates, this particular vector field v:


the_vector_v


In the Cartesian coordinate system, the curl formula is:


the_curl_formula_in_Cartesian_coordinates


Identify the vector components v1, v2 and v3:


the_components_of_v


Evaluating all the required partial derivatives:


the_partial_derivatives


Substituting into the curl formula:


substituting_into_the_curl_formula


Simplifying:


simplifying


As expected, the circulation points along the -z direction and we see the magnitude is 2.

The Curl in Cylindrical Coordinates

Find the curl of


the_vector_v_in_cylindrical_coordinate_system


(This is the same v but expressed in cylindrical coordinates.)

Write the curl formula:


the_curl_formula_in_cylindrical_coordinates


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