Subject: Calculus

# Divergence Theorem

Let E be a simple solid region and let S be the boundary surface of E, given with positive (outward) orientation. Let F be a vector field whose component functions have continuous partial derivatives on an open region that contains E. Then

\int_{S} F\cdot dS=\int \int \int_{E} (\nabla \cdot F )dV

## Discussion

**Divergence Theorem** is somewhat similar with Green's? and Stokes' Theorem in the same way that it relates the integral of a derivative of a certain function over a region to the integral of the original function over the boundary of the region. From the equation shown above, it tells us that the flux of F across the boundary surface of E is equal to the triple integral of the divergence of F over E.

To further simplify this theorem, let us first define the divergence of F denoted as ∇\cdot F which is define as follows:

Please take note that the dot product of a vector with another vector results to a scalar and thus, the divergence of F is a scalar.

Take for instance the figure above. r here has the components r=x\mathbf{\hat{x}} + y\mathbf{\hat{y}}+z \mathbf{\hat{z}}. If we take the divergence of F, we will have

Now let us apply Divergence Theorem on real calculus problem.

### Example

Find the flux of the vector field F(x, y, z) = (z\mathbf{\hat{i}}+ y\mathbf{\hat{j}}+ x\mathbf{\hat{k}} ) over the unit sphere x^2 + y^2 + z^2 = 1.

**Solution.**

To solve this, let us compute first the divergence of F:

The unit sphere S is the boundary of the unit ball B given by x^2 + y^2 + z^2 \leq 1. Thus, the Divergence Theorem gives the flux as

**NEXT TOPIC**: Stokes' theorem