How to Calculate Divergence and Curl

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Updated: January 28, 2017

In vector calculus, divergence and curl are two important types of operators used on vector fields. Because vector fields are ubiquitous, these two operators are widely applicable to the physical sciences.

Part One of Two:

  1. 1
    Understand what divergence is. Divergence is a measure of source or sink at a particular point. – In other words, how much is flowing into or out of a point. Hence, it is only defined for vector fields and outputs a scalar. Below is an example of a field with a positive divergence.
  2. 2
    Take the dot product of the partial derivatives with the components of , then sum the results. This applies for vector fields defined in Cartesian coordinates only.
  3. 3
    Use the formulas below as a reference. If the vector field is given in cylindrical or spherical coordinates (where is the polar angle), then the divergence does not have a simple form.
  4. 4
    Calculate the divergence of the following function.
    • As you can see, we have mapped from a vector field to a scalar field.

Part Two of Two:

  1. 1
    Understand what curl is. The curl, defined for vector fields, is, intuitively, the amount of circulation at any point. The operator outputs another vector field. A whirlpool in real life consists of water acting like a vector field with a nonzero curl. Above is an example of a field with negative curl (because it's rotating clockwise).
    • The curl is recognized by or , where the times symbol signifies the similarity of taking a cross product.
  2. 2
    Set up the determinant. The curl of a function is similar to the cross product of two vectors, hence why the curl operator is denoted with a As before, this mnemonic only works if is defined in Cartesian coordinates.
  3. 3
    Find the determinant of the matrix. Below, we do it by cofactor expansion (expansion by minors).
  4. 4
    Use the formulas below as a reference. The curl does not have a simple form if is in cylindrical or spherical coordinates.

  5. 5
    Calculate the curl of the following function.
  6. 6
    Set up the determinant.
  7. 7
    Calculate the determinant.
  8. 8
    Arrive at the answer.
    • Note that we have mapped to another vector field.

Community Q&A

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  • Question
    What is "curl vector"?
    Community Answer
    • Curl vector accounts for the rotatory effect a field produces. Consider a whirlpool: it's a fluid flow field with rotation. Its curl quantitatively represents the circulation of this field per unit area. (In simple terms, how much rotative effect per unit area a paddle wheel would feel when it's placed in this field.) A field with zero curl means a field with no rotation. Curl is a vector quantity as rotation must be represented with a vector (clockwise and anti-clockwise modes). By a simple analysis, it can be shown that for any field, F the curl can be completely represented as "curl(F)=nabla X F." (Nabla is the vector differential operator.)
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  • There are several identities that are useful and worth memorizing. Here is a partial list of the more important ones, the second derivatives. You can prove them by direct computation.
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  • The Laplacian is a compound operator and is defined as the divergence of the gradient We write out the Laplacian in Cartesian coordinates below. The operator is especially useful in physics and engineering. For example, the potential of an electrostatic field can be described using Poisson's equation, which involves the Laplacian.
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Updated: January 28, 2017
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Categories: Featured Articles | Calculus

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