Curl and Divergence

The divergence and curl are two derivatives of a vector field F = ⟨P, Q, R⟩. The divergence, div F = ∇ · F = P_x + Q_y + R_z, is a number that measures how much F spreads out from a point. The curl, curl F = ∇ × F, is a vector that measures how much F spins around a point, and about which axis.

Before this lesson: Vector Fields, Partial Derivatives, Cross Product and Green’s Theorem

What divergence and curl measure

Picture F as the velocity of a fluid. Draw a tiny sphere around a point. If more fluid flows out of the sphere than in, the point acts like a source and the divergence is positive. If more flows in, it acts like a sink and the divergence is negative. If the divergence is 0 everywhere, the fluid is incompressible.

Now put a tiny paddle wheel at the point. If the flow makes it spin, the curl is not zero. The curl vector points along the wheel’s axis (by the right-hand rule), and its length is twice the wheel’s angular speed.

Divergence and curl formulas

Treat ∇ = ⟨∂/∂x, ∂/∂y, ∂/∂z⟩ as a vector of derivatives. The divergence is its dot product with F, and the curl is its cross product with F.

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

\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} = \Big\langle R_y - Q_z,\; P_z - R_x,\; Q_x - P_y \Big\rangle

For a field in the plane, F = ⟨P, Q⟩, treat it as ⟨P, Q, 0⟩. Then div F = P_x + Q_y and curl F = ⟨0, 0, Q_x − P_y⟩, the same Q_x − P_y as in Green’s theorem.

Three facts to know

• curl(∇f) = 0 for every function f with continuous second partial derivatives.
• div(curl F) = 0 for every field F with continuous second partial derivatives.
• On all of space (or any region without holes), F is conservative exactly when curl F = 0.

Curl and divergence examples

Example 1: Divergence of a field in space

Find div F for F(x, y, z) = ⟨x²y, yz, xz²⟩, and evaluate it at (1, 2, −1).

P_x = \dfrac{\partial}{\partial x}(x^2y) = 2xy

Q_y = \dfrac{\partial}{\partial y}(yz) = z

R_z = \dfrac{\partial}{\partial z}(xz^2) = 2xz

\operatorname{div} \mathbf{F} = 2xy + z + 2xz

2(1)(2) + (-1) + 2(1)(-1) = 4 - 1 - 2 = 1

Answer: div F = 2xy + z + 2xz, and div F(1, 2, −1) = 1. The point acts like a weak source.

Example 2: Curl of the same field

Find curl F for F = ⟨x²y, yz, xz²⟩ and evaluate it at (1, 2, −1).

R_y - Q_z = 0 - y = -y

P_z - R_x = 0 - z^2 = -z^2

Q_x - P_y = 0 - x^2 = -x^2

\operatorname{curl} \mathbf{F} = \langle -y, -z^2, -x^2 \rangle

\operatorname{curl} \mathbf{F}(1, 2, -1) = \langle -2, -1, -1 \rangle

Answer: curl F = ⟨−y, −z², −x²⟩; at (1, 2, −1) it is ⟨−2, −1, −1⟩. Since the curl is not 0, F is not conservative.

Example 3: A source and a whirlpool

Find the divergence and curl of F = ⟨x, y⟩ and of G = ⟨−y, x⟩.

\operatorname{div} \mathbf{F} = 1 + 1 = 2

\operatorname{curl} \mathbf{F} = \langle 0, 0, 0 - 0 \rangle = \mathbf{0}

\operatorname{div} \mathbf{G} = 0 + 0 = 0

\operatorname{curl} \mathbf{G} = \langle 0, 0, 1 - (-1) \rangle = \langle 0, 0, 2 \rangle

Answer: F spreads out (div 2) without spinning (curl 0). G spins counterclockwise about the z-axis (curl ⟨0, 0, 2⟩) without spreading (div 0).

Example 4: Use the curl to test for a conservative field

Show that F = ⟨yz, xz + 2y, xy⟩ is conservative and find a potential function.

R_y - Q_z = x - x = 0

P_z - R_x = y - y = 0

Q_x - P_y = z - z = 0

f = \int yz\,dx = xyz + g(y, z)

xz + g_y = xz + 2y \Rightarrow g = y^2 + h(z)

xy + h'(z) = xy \Rightarrow h(z) = 0

Answer: f(x, y, z) = xyz + y². Check: ∇f = ⟨yz, xz + 2y, xy⟩. See conservative vector fields for more on potential functions.

Example 5: Stirring water in a tank

Water in a tank moves with velocity v = ⟨2x − y, x + y, −3z⟩ centimeters per second. Is the flow incompressible? How fast does a tiny paddle wheel with a vertical axis spin?

\operatorname{div} \mathbf{v} = 2 + 1 + (-3)

= 0

R_y - Q_z = 0 - 0 = 0, \quad P_z - R_x = 0 - 0 = 0

Q_x - P_y = 1 - (-1) = 2

\operatorname{curl} \mathbf{v} = \langle 0, 0, 2 \rangle

\omega = \dfrac{|\operatorname{curl} \mathbf{v}|}{2} = 1

Answer: yes, the flow is incompressible (div v = 0). The paddle wheel spins counterclockwise, seen from above, at 1 radian per second, at every point in the tank.

Common mistakes

Related methods

The Laplacian

The divergence of a gradient is the Laplacian: div(∇f) = f_xx + f_yy + f_zz, written ∇²f. Functions with ∇²f = 0 are called harmonic. They describe steady temperatures and electric potentials.

Divergence and curl in the big theorems

Green’s theorem uses the k-component of the curl. Stokes’ theorem uses the full curl on a surface, and the divergence theorem uses the divergence in a solid. Both appear at the end of this unit.

Practice problems

1. Find div F for F = ⟨3x, −2y, 5z⟩.
2. Find div F for F = ⟨xy, yz, zx⟩, and evaluate it at (1, 2, 3).
3. Find curl F for F = ⟨y, z, x⟩.
4. Find curl F for F = ⟨x², y², z²⟩.
5. For the plane field F = ⟨x² − y, xy⟩, find div F and the k-component of curl F.
6. Is F = ⟨2xy, x² + z, y⟩ conservative? If so, find a potential function.
7. For F = ⟨xyz, x², yz⟩, find curl F and check that div(curl F) = 0.
8. Let f = x²y³z. Find ∇f and check that curl(∇f) = 0.
9. Find the number a so that F = ⟨ax + y, x − 3y, 2z⟩ has divergence 0.
10. On a spinning turntable, the velocity of the surface is v = ⟨−3y, 3x, 0⟩ m/s. Find curl v, div v and the angular speed of the turntable.

\operatorname{div} \mathbf{F} = 3 + (-2) + 5 = 6

\operatorname{div} \mathbf{F} = y + z + x

2 + 3 + 1 = 6

R_y - Q_z = 0 - 1 = -1

P_z - R_x = 0 - 1 = -1

Q_x - P_y = 0 - 1 = -1

R_y - Q_z = 0, \quad P_z - R_x = 0, \quad Q_x - P_y = 0

\operatorname{div} \mathbf{F} = 2x + x = 3x

Q_x - P_y = y - (-1) = y + 1

R_y - Q_z = 1 - 1 = 0, \quad P_z - R_x = 0 - 0 = 0, \quad Q_x - P_y = 2x - 2x = 0

f = x^2y + g(y, z), \quad x^2 + g_y = x^2 + z \Rightarrow g = yz + h(z)

y + h'(z) = y \Rightarrow h = 0

\operatorname{curl} \mathbf{F} = \langle z - 0, xy - 0, 2x - xz \rangle

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

\nabla f = \langle 2xy^3z, 3x^2y^2z, x^2y^3 \rangle

R_y - Q_z = 3x^2y^2 - 3x^2y^2 = 0

P_z - R_x = 2xy^3 - 2xy^3 = 0, \quad Q_x - P_y = 6xy^2z - 6xy^2z = 0

\operatorname{div} \mathbf{F} = a - 3 + 2 = a - 1

a - 1 = 0 \Rightarrow a = 1

\operatorname{curl} \mathbf{v} = \langle 0, 0, 3 - (-3) \rangle = \langle 0, 0, 6 \rangle

\operatorname{div} \mathbf{v} = 0 + 0 + 0 = 0

\omega = \dfrac{6}{2} = 3

Curl and divergence FAQ

What does the divergence of a vector field tell you?

It measures the net outflow per unit volume at a point. Positive divergence means a source, negative means a sink, and zero means the flow is incompressible there.

What does the curl of a vector field tell you?

It measures rotation. The curl points along the axis a tiny paddle wheel would spin around, and its length is twice the wheel’s angular speed.

Is the curl a vector or a scalar?

The curl is a vector, and the divergence is a scalar. For a field in the plane, people often call the single number Q_x − P_y the scalar curl, but it is really the k-component of the curl.

How do you know if a vector field is conservative using the curl?

If F has continuous partial derivatives on all of space (or a region without holes) and curl F = 0, then F is conservative. If curl F is not 0 anywhere, F is not conservative.

Summary

• div F = ∇ · F = P_x + Q_y + R_z, a scalar that measures outflow.
• curl F = ∇ × F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, a vector that measures rotation.
• curl(∇f) = 0 and div(curl F) = 0.
• On a region without holes, curl F = 0 means F is conservative.
• div F = 0 means incompressible; curl F = 0 means irrotational.

Previous lesson: Green’s Theorem

Next lesson: Parametric Surfaces

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