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:

\nabla \cdot F=\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} +\frac{\partial F_z}{\partial z}

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

\nabla \cdot F=\left (\frac{\partial }{\partial x}\mathbf{\hat{x}} + \frac{\partial }{\partial y}\mathbf{\hat{y}} +\frac{\partial }{\partial z}\mathbf{\hat{z}}\right )\cdot \left ( x\mathbf{\hat{x}}+y\mathbf{\hat{y}}+z\mathbf{\hat{z}}\right )
=\frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} +\frac{\partial z}{\partial z}
=1+1+1=3
.

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:

\nabla \cdot F=\left (\frac{\partial }{\partial x}\mathbf{\hat{i}} + \frac{\partial }{\partial y}\mathbf{\hat{j}} +\frac{\partial }{\partial z}\mathbf{\hat{k}}\right )\cdot \left ( z\mathbf{\hat{i}}+y\mathbf{\hat{j}}+x\mathbf{\hat{k}}\right )
=\frac{\partial z}{\partial x} + \frac{\partial y}{\partial y} +\frac{\partial x}{\partial z}
=0+1+0=1
.

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

\int_{S} F\cdot dS=\int \int \int_{B} (\nabla \cdot F )dV =\int \int \int_{E}1dV

=V(B)=\frac{4}{3}\pi (1)^2=\frac{4\pi}{3}

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