 # 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 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 : The theorem gets its name from the divergence of a vector field F = Mi + Nj + Pk. 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 = 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: ## References

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

CITE THIS AS:
Stephanie Glen. "Divergence Theorem: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/divergence-theorem/
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