Math Core

A vector field can do two basic things near a point: swirl around it or spread out from it. The curl measures the swirl and the divergence measures the spreading. Both are computed from partial derivatives using the "del" operator $\nabla$ , and they are the key ingredients in the two big theorems still to come, Stokes' theorem and the divergence theorem.

The del operator

Write $\nabla$ as a vector of differentiation operators:

\nabla = \left\langle \frac{\partial}{\partial x},\ \frac{\partial}{\partial y},\ \frac{\partial}{\partial z}\right\rangle.

Applied to a scalar function, $\nabla f$ is the gradient. Applied to a vector field $\mathbf{F} = \langle P, Q, R\rangle$ with a cross product or a dot product, it gives the curl and the divergence.

Curl

Definition

Curl

The curl of $\mathbf{F} = \langle P, Q, R\rangle$ is the vector field

\operatorname{curl}\mathbf{F} = \nabla\times\mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} = \left\langle R_y - Q_z,\ P_z - R_x,\ Q_x - P_y\right\rangle.

Meaning. Imagine $\mathbf{F}$ is the velocity of a fluid and you drop a tiny paddle wheel into it. The wheel spins fastest when its axle points along $\operatorname{curl}\mathbf{F}$ , and the length of the curl is proportional to how fast it spins (twice the angular speed, in fact). If $\operatorname{curl}\mathbf{F} = \mathbf{0}$ , the field is irrotational. Notice that the third component, $Q_x - P_y$ , is exactly the integrand in Green's theorem.

Worked example: Computing curl and divergence

Let $\mathbf{F} = \langle xy,\ yz^2,\ x^2z\rangle$ . Then $P = xy$ , $Q = yz^2$ , $R = x^2z$ , and

\operatorname{curl}\mathbf{F} = \left\langle 0 - 2yz,\ 0 - 2xz,\ 0 - x\right\rangle = \langle -2yz,\ -2xz,\ -x\rangle.

The divergence (defined below) is $\dfrac{\partial}{\partial x}(xy) + \dfrac{\partial}{\partial y}(yz^2) + \dfrac{\partial}{\partial z}(x^2z) = y + z^2 + x^2$ .

Curl and conservative fields

Two facts connect curl to the previous lessons.

Curl detects conservative fields

For any $f$ with continuous second partial derivatives, $\operatorname{curl}(\nabla f) = \mathbf{0}$ . So every conservative field is irrotational.
Conversely, if $\mathbf{F}$ has continuous partial derivatives on all of $\mathbb{R}^3$ (or any simply connected region) and $\operatorname{curl}\mathbf{F} = \mathbf{0}$ , then $\mathbf{F}$ is conservative.

Fact 1 is Clairaut's Theorem: for instance, the first component of $\operatorname{curl}(\nabla f)$ is $f_{zy} - f_{yz} = 0$ . Fact 2 is the three-dimensional version of the $P_y = Q_x$ test.

Worked example: Is it conservative?

Let $\mathbf{F} = \langle y^2z^3,\ 2xyz^3,\ 3xy^2z^2\rangle$ . Then

\begin{aligned} R_y - Q_z &= 6xyz^2 - 6xyz^2 = 0, \\ P_z - R_x &= 3y^2z^2 - 3y^2z^2 = 0, \\ Q_x - P_y &= 2yz^3 - 2yz^3 = 0. \end{aligned}

So $\operatorname{curl}\mathbf{F} = \mathbf{0}$ on $\mathbb{R}^3$ and $\mathbf{F}$ is conservative. Integrating as in the previous lesson gives the potential $f = xy^2z^3$ .

Divergence

Definition

Divergence

The divergence of $\mathbf{F} = \langle P, Q, R\rangle$ is the scalar function

\operatorname{div}\mathbf{F} = \nabla\cdot\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}.

In the plane, $\operatorname{div}\langle P, Q\rangle = P_x + Q_y$ .

Meaning. If $\mathbf{F}$ is a fluid velocity, $\operatorname{div}\mathbf{F}(P)$ is the net rate at which fluid flows out of a tiny box around $P$ , per unit volume. Positive divergence means a source (fluid spreading out), negative means a sink, and $\operatorname{div}\mathbf{F} = 0$ everywhere means the fluid is incompressible. For example, $\langle x, y, z\rangle$ has divergence $3$ : it pushes outward everywhere.

A second identity pairs with $\operatorname{curl}(\nabla f) = \mathbf{0}$ :

\operatorname{div}(\operatorname{curl}\mathbf{F}) = 0

for any $\mathbf{F}$ with continuous second partials. (Expand it and the mixed partials cancel in pairs.) Also, $\operatorname{div}(\nabla f) = f_{xx} + f_{yy} + f_{zz}$ is called the Laplacian of $f$ , written $\nabla^2 f$ .

Worked example: A field that is not a curl

Is there a vector field $\mathbf{G}$ with $\operatorname{curl}\mathbf{G} = \langle x,\ y,\ z\rangle$ ?

If there were, then $\operatorname{div}(\operatorname{curl}\mathbf{G})$ would be $0$ . But $\operatorname{div}\langle x, y, z\rangle = 1 + 1 + 1 = 3 \ne 0$ . So no such $\mathbf{G}$ exists.

Common mistake

Curl is a vector and divergence is a scalar. The expression $\operatorname{curl}(\operatorname{div}\mathbf{F})$ is meaningless, because you can't take the curl of a scalar. Also keep the curl's middle component straight: it is $P_z - R_x$ , not $R_x - P_z$ . The cofactor expansion's minus sign flips it.

Green's theorem in vector form

Curl and divergence let you write Green's theorem in two ways for a plane field $\mathbf{F} = \langle P, Q\rangle$ and a positively oriented curve $C$ around $D$ .

Two forms of Green's theorem

Circulation form: $\displaystyle\oint_C \mathbf{F}\cdot\mathbf{T}\,ds = \iint_D (\operatorname{curl}\mathbf{F})\cdot\mathbf{k}\,dA = \iint_D (Q_x - P_y)\,dA$ .

Flux form: $\displaystyle\oint_C \mathbf{F}\cdot\mathbf{n}\,ds = \iint_D \operatorname{div}\mathbf{F}\,dA = \iint_D (P_x + Q_y)\,dA$ ,

where $\mathbf{n}$ is the outward unit normal to $C$ .

The flux form says the net outflow across the boundary equals the total of all the sources inside. Stokes' theorem and the divergence theorem are these two statements lifted into three dimensions.

Worked example: Outward flux across a circle

Find the outward flux of $\mathbf{F} = \langle x, y\rangle$ across the unit circle.

$\operatorname{div}\mathbf{F} = 1 + 1 = 2$ , so the flux is $\iint_D 2\,dA = 2\pi$ . Check directly: on the unit circle the outward normal is $\mathbf{n} = \langle x, y\rangle$ , so $\mathbf{F}\cdot\mathbf{n} = x^2 + y^2 = 1$ , and the flux is the length $2\pi$ .

Practice

Practice 1

Find $\operatorname{div}\mathbf{F}$ for $\mathbf{F} = \langle x^2y,\ yz,\ xz^2\rangle$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 2

Find $\operatorname{curl}\mathbf{F}$ for $\mathbf{F} = \langle y,\ z,\ x\rangle$ .

Enter a point like (2, -3)

Practice 3

Let $\mathbf{F} = \langle x^2z,\ yz^2,\ xy\rangle$ . Find $\operatorname{curl}\mathbf{F}$ at the point $(1, 2, 1)$ .

Enter a point like (2, -3)

Practice 4

Find $\operatorname{div}\mathbf{F}$ for $\mathbf{F} = \langle e^x\sin y,\ e^x\cos y,\ z\rangle$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Assume $f$ and $\mathbf{F}$ have continuous second partial derivatives. Which expression is always equal to $0$ ?

Practice 6

Find the constant $a$ that makes $\mathbf{F} = \langle axy + z,\ x^2,\ x\rangle$ conservative on $\mathbb{R}^3$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

$\mathbf{F} = \langle 2xyz,\ x^2z,\ x^2y + 1\rangle$ has zero curl. Evaluate $\displaystyle\int_C \mathbf{F}\cdot d\mathbf{r}$ along any path from $(0, 0, 0)$ to $(1, 2, 3)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Use the flux form of Green's theorem to find the outward flux of $\mathbf{F} = \langle x^3,\ y^3\rangle$ across the unit circle.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.