Divergence and Curl: Step-by-Step Guide
For a vector field F = (P, Q, R), the divergence div F is a scalar that measures net outflow per unit volume, and the curl curl F is a vector that measures rotation per unit area. Divergence is the dot product of del with F; curl is the cross product of del with F.
Math Racer skill guide · Calculus 3 · College
How to solve it, step by step
- Write F in component form. Identify the components P, Q, R of F = (P(x,y,z), Q(x,y,z), R(x,y,z)).\[\mathbf{F} = \langle P, Q, R \rangle\]
- Compute divergence. Add the partial of P with respect to x, Q with respect to y, and R with respect to z.\[\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]
- Set up the curl as a determinant. Use the symbolic 3x3 determinant with i, j, k in the first row, partials in the second row, and P, Q, R in the third.\[\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ P & Q & R \end{vmatrix}\]
- Expand the determinant. curl F has components (R_y - Q_z, P_z - R_x, Q_x - P_y).\[\nabla\times\mathbf{F} = \langle R_y - Q_z,\; P_z - R_x,\; Q_x - P_y \rangle\]
- Interpret the results. div F = 0 means F is incompressible; curl F = 0 means F is irrotational (and locally a gradient).\[\nabla\cdot\mathbf{F}=0,\; \nabla\times\mathbf{F}=\mathbf{0}\]
Worked example
For F = (x*y, y*z, z*x), find div F and curl F.
Step 1: Identify P = xy, Q = yz, R = zx.
Step 2: Compute divergence. P_x = y, Q_y = z, R_z = x. So div F = y + z + x.
Step 3: Compute partials needed for curl. R_y = 0, Q_z = y, P_z = 0, R_x = z, Q_x = 0, P_y = x.
Step 4: Assemble curl F = (R_y - Q_z, P_z - R_x, Q_x - P_y) = (0 - y, 0 - z, 0 - x) = (-y, -z, -x).
Step 5: Result: div F = x + y + z and curl F = (-y, -z, -x).
Verification: Check div(curl F) = -y_x - z_y - x_z = 0. This always holds (a fundamental identity), so our curl is consistent.
Tips and common mistakes
- Divergence is a scalar; curl is a vector. Don't confuse the two.
- Always use the determinant formula for curl - it prevents sign errors.
- Identity: div(curl F) = 0 always; use this to check curl computations.
- Identity: curl(grad f) = 0 always; if F = grad f, then curl F = 0.
- If curl F = 0 on a simply connected region, F is conservative and equals grad f for some f.
- div F at a point is the rate of fluid expansion there (positive = source, negative = sink).
- For 2D vector fields F = (P, Q), curl F is sometimes written as the scalar Q_x - P_y.
- Watch the order of variables in partials: R_y means d R / d y, not d R / d r.