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Divergence Theorem: Definition, Examples

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The divergence theorem (sometimes called Gauss’s or Ostrogradsky’s theorem) is a connection between triple (volume) integrals and surface integrals. It states that under certain conditions, some volume integrals are equal to surface integrals. This can lead to simpler integration.

As a simple example, in order to perform integration on a shape like a pyramid, you would need four surface integrals: one for each side. But it’s possible, if the shape meets a few basic requirements, to use one triple integral instead.

Formal Definition of the Divergence Theorem

The theorem is valid for a closed surface S, meaning that S forms the complete boundary of the solid. Closed surfaces include regions bounded by cubes, ellipsoids, spheres, tetrahedrons, or combinations of those surfaces.

Formal Definition of the Divergence Theorem

the divergence theorem

The two parts of this solid region Q are oriented by one upward unit normal vector N (S1) and one downward one (S2).

The above image shows an example of the type of shape the divergence theorem covers; A solid region, which we’ll call Q, bounded by a closed surface S and oriented by one or more unit normal vectors directed outward from Q.

If F is a vector field with component functions that have continuous first partial derivatives in Q, then [1]:
formal definition of divergence theorem

The theorem gets its name from the divergence of a vector field F = Mi + Nj + Pk. This is defined as:
divergence of a vector field

Where ∂ is a partial derivative.


divergence theorem example

The above image is bounded by four planes: the coordinate planes x, y and the plane 2x + 2y + z = 6 with
F = xi + y2j + zk.

Instead of evaluating a surface integral for the four different planes, we can use a triple integral instead.

We need to find div F before we can solve the triple integral. We have:

  • = 1 + 2y + 1
  • = 2 + 2y.

I used Symbolab’s Triple Integrals Calculator to evaluate the triple integral. The solution is:
symbolab solution for triple integral


[1] Larson, R. & Edwards, B. (2016). Calculus, 10th Edition. Cengage Learning.

Stephanie Glen. "Divergence Theorem: Definition, Examples" From Calculus for the rest of us!

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