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 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

Fis a vector field with component functions that have continuous first partial derivatives in Q, then [1]:

The theorem gets its name from the *divergence* of a vector field **F** = M**i** + N**j** + P**k**. This is defined as:

Where ∂ is a partial derivative.

## Example

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

**F** = x**i** + y^{2}**j** + z**k**.

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:

## References

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

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