Vector calculus

A number at every point

• A voltage probe reports one number at each point in space: a whole landscape of values. Later, Maxwell's equations will be statements about that landscape's shape.
• Three operators describe the shape: which way is uphill, where the field flows out, and where it swirls.

Fields and their three operators

• A scalar field gives a number at each point (potential $V$); a vector field gives an arrow (electric field $\mathbf{E}$).
• Gradient 梯度 turns a scalar field into a vector field; divergence 散度 turns a vector field into a scalar; curl 旋度 turns a vector field into another vector field.

Gradient points uphill

• $\nabla V = \left(\dfrac{\partial V}{\partial x}, \dfrac{\partial V}{\partial y}, \dfrac{\partial V}{\partial z}\right)$ points up the steepest slope of $V$. The electric field points downhill: $\mathbf{E} = -\nabla V$.
• $V = 3x^{2}y$ volts gives $\nabla V = (6xy,\ 3x^{2})$. At $(1, 2)$: $\nabla V = (12, 3)$, so $\mathbf{E} = -(12, 3)\,\dfrac{\text{V}}{\text{m}}$.

Check the gradient with a small step

• Take a small step $\boldsymbol{\delta}$ from the point. The potential changes by about $\nabla V \cdot \boldsymbol{\delta}$, which is $-\mathbf{E} \cdot \boldsymbol{\delta}$.
• From $(1, 2)$, step $\boldsymbol{\delta} = (0.01, 0)$: $\Delta V \approx 12 \times 0.01 = 0.12\,\text{V}$, and directly $3(1.01)^{2}(2) - 6 = 0.1206\,\text{V}$. The field predicts the drop.

Divergence: net outflow

• $\nabla \cdot \mathbf{F} = \dfrac{\partial F_{x}}{\partial x} + \dfrac{\partial F_{y}}{\partial y} + \dfrac{\partial F_{z}}{\partial z}$ is the net outflow per unit volume. For an electric field it measures the charge density there.
• $\mathbf{F} = (x^{2},\ xy,\ 3x^{2}z)$: $\nabla \cdot \mathbf{F} = 2x + x + 3x^{2}$, which is 6 at $(1, 1, 1)$: a net source.

Curl: circulation

• The $z$-component of curl is $\dfrac{\partial F_{y}}{\partial x} - \dfrac{\partial F_{x}}{\partial y}$: the circulation per unit area. Electrostatic fields have zero curl; induction is where it is not zero.
• $\mathbf{G} = (-y,\ x,\ 0)$ gives $1 - (-1) = 2$ everywhere. This is a rigid body spinning at 1 rad/s: curl is twice the angular velocity.

Three common slips

Recap