The Divergence of a Vector Field Examples 2

The Divergence of a Vector Field Examples 2

Recall from The Divergence of a Vector Field page that if $\mathbf{F} (x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a vector field on $\mathbb{R}^3$ then the divergence of $\mathbf{F}$ is given by the following formula:

(1)
\begin{align} \quad \mathrm{div} (\mathbf{F}) = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \end{align}

We will now look at some more examples of computing the divergence of a vector field.

Example 1

Find the divergence of the vector field $\mathbf{F} (x, y, z) = \frac{x^2}{x^2 + 2y^2} \vec{i} + x \sin y \vec{j} + \frac{1 - e^z}{y} \vec{k}$.

Applying the formula for the divergence of a vector field directly by calculating the necessary partial derivatives and we get that:

(2)
\begin{align} \quad \mathrm{div} (\mathbf{F}) = \frac{\partial}{\partial x} \left ( \frac{x^2}{x^2 + 2y^2} \right ) + \frac{\partial}{\partial y} \left ( x \sin y \right ) + \frac{\partial}{\partial z} \left ( \frac{1 - e^z}{y} \right ) \\ \quad \mathrm{div} (\mathbf{F}) = \frac{(x^2 + 2y^2)(2x) - x^2(2x)}{(x^2 + 2y^2)^2} + x \cos y - \frac{e^z}{y} \\ \quad \mathrm{div} (\mathbf{F}) = \frac{4xy^2}{(x^2 + 2y^2)^2} + x \cos y - \frac{e^z}{y} \end{align}

Example 2

Find the divergence of the vector field $\mathbf{F} (x, y, z) = \frac{x^2 + y^2 + z^2}{xyz} \vec{i} + \frac{\cos(x)\sin(y)\tan(z)}{xyz} \vec{j} + \sin(2x + eyz) \vec{k}$.

Using the formula above for the divergence of a vector field and we get that:

(3)
\begin{align} \quad \mathrm{div} (\mathbf{F}) = \frac{\partial}{\partial x} \left ( \frac{x^2 + y^2 + z^2}{xyz} \right ) + \frac{\partial}{\partial y} \left ( \frac{\cos(x)\sin(y)\tan(z)}{xyz} \right ) + \frac{\partial}{\partial z} \left ( \sin(2x + eyz) \right ) \\ \quad \mathrm{div} (\mathbf{F}) = \frac{(xyz)(2x) - (x^2 + y^2 + z^2)(yz)}{x^2y^2z^2} + \frac{(xyz)[\cos (x) \cos (y) \tan (z)] - [\cos(x) \sin (y) \tan (z)] (xz)}{x^2 + y^2 + z^2} + ey \cos (2x + eyz) \end{align}