A vector field has many partial derivatives: $P_x$, $P_y$, $Q_x$, and so on. Two combinations of them have a clear meaning and show up in every big theorem of this course. Divergence measures how much a field spreads out from a point, and curl measures how much it swirls. Both are written with the symbol $\nabla = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right\rangle$, used like a vector whose "components" are derivatives. You need partial derivatives and the cross product.
Key idea
The curl formula comes from the determinant pattern of a cross product: $$\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ P & Q & R \end{vmatrix},$$ where "multiplying" $\partial_y$ by $R$ means taking $R_y$. As with any ${}3 \times 3$ determinant, the middle component carries a minus sign: $-(R_x - P_z) = P_z - R_x$.
Why it works
$$\nabla \times (\nabla f) = \langle f_{zy} - f_{yz},\ f_{xz} - f_{zx},\ f_{yx} - f_{xy} \rangle = \mathbf{0}\text{:}$$ gradient fields have zero curl.
$$\nabla \cdot (\nabla \times \mathbf{F}) = (R_y - Q_z)_x + (P_z - R_x)_y + (Q_x - P_y)_z = R_{yx} - Q_{zx} + P_{zy} - R_{xy} + Q_{xz} - P_{yz} = 0\text{:}$$ the terms cancel in pairs.
So a field with nonzero curl is not a gradient field, and a field with nonzero divergence is not a curl.
| Operation | Takes | Gives | Measures |
|---|---|---|---|
| gradient $\nabla f$ | a scalar function | a vector field | steepest increase |
| divergence $\nabla \cdot \mathbf{F}$ | a vector field | a scalar function | spreading out (sources) |
| curl $\nabla \times \mathbf{F}$ | a vector field | a vector field | swirling (rotation) |
Worked examples
Worked example: A divergence
- $P_x = 2xy$, $Q_y = z$, $R_z = 2xz$.
- $\nabla \cdot \mathbf{F} = 2xy + z + 2xz$, a single number at each point.
Answer: ${}2xy + z + 2xz$
Worked example: A curl that vanishes
- $R_y - Q_z = x - x = 0$.
- $P_z - R_x = y - y = 0$.
- $Q_x - P_y = z - z = 0$.
- $\nabla \times \mathbf{F} = \mathbf{0}$, as expected: $\mathbf{F} = \nabla(xyz)$.
Answer: $\mathbf{0}$
Worked example: Both at once
- Curl: $R_y - Q_z = 0$, $P_z - R_x = 0$, $Q_x - P_y = 1 - (-1) = 2$, so $\nabla \times \mathbf{F} = \langle 0, 0, 2 \rangle$.
- Divergence: ${}0 + 0 + 2z = 2z$.
- The field spins about the $z$-axis everywhere and spreads out only where $z > 0$.
Answer: curl $\langle 0, 0, 2 \rangle$, divergence ${}2z$
Common mistake