Gradient, Divergence, and Curl

Source lecture(s): PC604 Lec1-2, PHY621 Lec1

Intuition

Gradient points uphill; divergence measures source strength; curl measures local circulation.

Formal Definition

For scalar field $\phi$ and vector field $\vec{A}$: $\nabla\phi$, $\nabla\cdot\vec{A}$, $\nabla\times\vec{A}$.

Mathematical Formulation

\[\nabla\phi=\hat{e}_i\frac{\partial\phi}{\partial x_i}$$ $$\nabla\cdot\vec{A}=\frac{\partial A_i}{\partial x_i}$$ $$\nabla\times\vec{A}=\epsilon_{ijk}\hat{e}_i\frac{\partial A_k}{\partial x_j}\]

Derivation

In curvilinear coordinates insert scale factors $h_i$. For Cartesian $h_i=1$, this collapses to the familiar forms.

Worked Example

For $\vec{A}=x\hat{i}+y\hat{j}+z\hat{k}$, $\nabla\cdot\vec{A}=3$. For $\vec{A}=-y\hat{i}+x\hat{j}$, $\nabla\times\vec{A}=2\hat{k}$.

Common Mistakes

Treating $\nabla$ as a regular vector when differentiating products.
Assuming zero divergence means no field variation.

Quiz

Q1. What does $\nabla\cdot\vec{A}=0$ mean physically?

Answer

No net source or sink in the field.

Q2. What does $\nabla\times\nabla\phi$ equal?