Divergence
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 6.5: “Divergence and Curl” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/6-5-divergence-and-curl |
| Textbook used in class | Stewart, Calculus, Section 16.5: “Curl and Divergence” (Examples 3, 4) |
Quick Reference
Divergence of $\mathbf{F} = \langle P, Q, R\rangle$: $$\text{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}.$$
Laplacian: $\text{div}(\nabla f) = \nabla^2 f = f_{xx}+f_{yy}+f_{zz}$.
Key identity: $\text{div}(\text{curl}\,\mathbf{F}) = 0$.
Motivation
Divergence measures the net outward flow of a vector field per unit volume at a point. Positive divergence: the field spreads out (like the field around a source -- a faucet in a fluid). Negative divergence: the field converges (like the field around a sink -- a drain). Zero divergence: the field is “incompressible” -- the same amount flows in as flows out locally.
The divergence theorem (Section 16.9) makes this precise: the total flux out of a closed surface equals the integral of divergence over the enclosed volume.
Key Concepts
1. Computing Divergence
Divergence is a scalar: add the three partial derivatives $P_x + Q_y + R_z$. This is the dot product $\nabla\cdot\mathbf{F}$ where $\nabla = \langle\partial/\partial x, \partial/\partial y, \partial/\partial z\rangle$.
2. Physical Interpretation
For a fluid velocity field $\mathbf{v}$, $\text{div}\,\mathbf{v}(P)$ measures the rate at which fluid is being produced (source) or absorbed (sink) at the point $P$ per unit volume. An incompressible fluid has $\text{div}\,\mathbf{v} = 0$ everywhere.
3. The Identity $\text{div}(\text{curl}\,\mathbf{F}) = \mathbf{0}$
For any smooth $\mathbf{F}$, $\text{div}(\text{curl}\,\mathbf{F}) = 0$. This follows directly from equality of mixed partial derivatives (Clairaut’s theorem). It means the curl of a field is always incompressible.
Worked Example
Compute $\text{div}\,\mathbf{F}$ for $\mathbf{F}(x,y,z) = \langle xy, e^{yz}, x^2z\rangle$. (Stewart 16.5, Example 3.)
$P = xy$: $P_x = y$.
$Q = e^{yz}$: $Q_y = ze^{yz}$.
$R = x^2 z$: $R_z = x^2$.
$$\text{div}\,\mathbf{F} = y + ze^{yz} + x^2.$$
divergence is a vector. Divergence is a SCALAR: $\text{div}\,\mathbf{F} = P_x + Q_y + R_z$. Curl is a vector (three components). The del operator $\nabla$ gives curl when crossed ($\nabla\times\mathbf{F}$, a vector) and divergence when dotted ($\nabla\cdot\mathbf{F}$, a scalar). The notations $\nabla\times$ and $\nabla\cdot$ indicate which operation produces which type of result.
Common Misconceptions
divergence is a vector, like curl.
This is the multiplicative-not-additive error. Divergence is the dot product $\nabla \cdot \mathbf{F} = P_x + Q_y + R_z$, which is a scalar. Curl is the cross product $\nabla \times \mathbf{F}$, which is a vector. The two operations are distinguished by whether $\nabla$ is dotted or crossed with $\mathbf{F}$: dotting collapses the three components into one number, while crossing produces a new three-component vector. A common sign of this error is writing $\operatorname{div}\mathbf{F} = \langle P_x, Q_y, R_z\rangle$ instead of $P_x + Q_y + R_z$.
Leveled Practice
Problem 1. Find $\text{div}\,\mathbf{F}$ for $\mathbf{F} = \langle x^2 z, -xy^2, yz^2\rangle$ and verify $\text{div}(\text{curl}\,\mathbf{G}) = 0$ for $\mathbf{G} = \langle xy, yz, xz\rangle$.
Show answer
$\text{div}\,\mathbf{F} = 2xz - 2xy + 2yz$.
For $\mathbf{G}$: $\text{curl}\,\mathbf{G} = \langle z_y - y_{yz}, x_{xz} - z_x, y_x - x_y\rangle$...
More concretely: $P=xy$, $Q=yz$, $R=xz$.
$\text{curl}\,\mathbf{G} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y\rangle = \langle 0-y, 0-z, 0-x\rangle = \langle -y, -z, -x\rangle$.
$\text{div}(\text{curl}\,\mathbf{G}) = \partial(-y)/\partial x + \partial(-z)/\partial y + \partial(-x)/\partial z = 0+0+0 = 0$. Confirmed.