Multivariable Calculus: Grad, Div, Curl
Vector fields, the three differential operators, and Stokes-type theorems.
For a scalar field $f: \mathbb R^n \to \mathbb R$, the gradient $\nabla f = (\partial_1 f, \ldots, \partial_n f)$ points in the direction of steepest ascent; $|\nabla f|$ is the rate of increase in that direction.
For a vector field $\mathbf F = (F_1, F_2, F_3)$ in $\mathbb R^3$:
$$\nabla \cdot \mathbf F = \partial_1 F_1 + \partial_2 F_2 + \partial_3 F_3 \quad (\text{divergence}),$$ $$\nabla \times \mathbf F = (\partial_2 F_3 - \partial_3 F_2,\; \partial_3 F_1 - \partial_1 F_3,\; \partial_1 F_2 - \partial_2 F_1) \quad (\text{curl}).$$The integral theorems unify:
- Gradient theorem: $\int_\gamma \nabla f \cdot d\mathbf r = f(\gamma(b)) - f(\gamma(a))$.
- Stokes: $\int_{\partial \Sigma} \mathbf F \cdot d\mathbf r = \int_\Sigma (\nabla \times \mathbf F) \cdot d\mathbf S$.
- Divergence theorem: $\int_{\partial V} \mathbf F \cdot d\mathbf S = \int_V \nabla \cdot \mathbf F \, dV$.
Each is "boundary integral = bulk integral of one derivative more" — special cases of the general Stokes theorem on differential forms.