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Finding the Divergence 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 we look at finding the divergence of vector field in three different coordinate systems. The same vector field expressed in each of the coordinate systems is used in the examples.

Find the Divergence of a Vector Field

Step 1: Identify the coordinate system.

One way to identify the coordinate system is to look at the unit vectors. If you see unit vectors with:

  • two different angles (typically, θ and φ), we are using spherical coordinates.
  • only one of the unit vectors referring to an angle, the coordinate system is cylindrical.
  • none of the unit vectors referencing an angle, we have the Cartesian coordinate system.

Step 2: Lookup (or derive) the divergence formula for the identified coordinate system.

The vector field is v. The symbol ∇ (called a ''nabla'') with a dot means to find the divergence of the vector field. The formula depends on the coordinate system.

  • Cartesian coordinate system:


divergence_cartesian


  • Cylindrical coordinate system:


divergence_cylindrical


  • Spherical coordinate system:


divergence_spherical


Step 3: Identify the vector components v1, v2 and v3.

The vector components are the terms multiplying the unit vectors. For example, given the following vector,


v_cartesian


the vector components are


cartesian_components


Step 4: Substitute the components into the divergence formula, evaluate and simplify.

The Final Solution in Each of the Coordinate Systems

Cartesian Coordinate System

As we've seen, the following vector in the Cartesian coordinate system,


v_cartesian


has vector components


cartesian_components


The divergence is


div_cartesian


The divergence of a vector field is a scalar quantity, and for this vector field, the divergence is 2.

Cylindrical Coordinate System

This same vector field expressed in the cylindrical coordinate system is


v_cylindrical


with components


cylindrical_components


and divergence


div_cylindrical


We get a result equal to 2 as expected since it's the divergence of the same vector field but expressed in a difference coordinate system. Usually, choosing a coordinate system which matches the symmetry of the problem will lead to simpler computations.

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