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:


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

  • 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

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,


The parameter values for the cylindrical coordinate system:


For the spherical coordinate system:


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

For the Cartesian coordinate system, substituting:


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


For the cylindrical coordinate system


Expanding the determinant:


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


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


Expanding the determinant:




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


The Final Results

The curl equation in each of our coordinate systems:

Cartesian Coordinate System


Cylindrical Coordinate System


Spherical Coordinate System


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

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:


In the Cartesian coordinate system, the curl formula is:


Identify the vector components v1, v2 and v3:


Evaluating all the required partial derivatives:


Substituting into the curl formula:




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


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

Write the curl formula:


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