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:
Cylindrical coordinate system:
Spherical coordinate system:
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,
the vector components are
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,
has vector components
The divergence is
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
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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