Work done by a force field. 4. Surface Integrals For surface $S$ with unit normal $\mathbfn$: $$\iint_S \mathbfF \cdot d\mathbfS = \iint_S \mathbfF \cdot \mathbfn , dS$$
A set $V$ with addition and scalar multiplication satisfying closure, associativity, commutativity, zero element, additive inverse, and distributivity. linear algebra and vector analysis pdf
| Theorem | Equation | Meaning | |---------|----------|---------| | | $\int_C \nabla f \cdot d\mathbfr = f(\mathbfr(b)) - f(\mathbfr(a))$ | Line integral of gradient = difference of potential | | Green's Theorem | $\oint_C (P,dx + Q,dy) = \iint_D \left( \frac\partial Q\partial x - \frac\partial P\partial y \right) dA$ | Relates line integral to double integral | | Divergence Theorem | $\iint_S \mathbfF \cdot d\mathbfS = \iiint_V (\nabla \cdot \mathbfF) , dV$ | Flux through closed surface = volume integral of divergence | | Stokes' Theorem | $\oint_C \mathbfF \cdot d\mathbfr = \iint_S (\nabla \times \mathbfF) \cdot d\mathbfS$ | Circulation = flux of curl | Part III: The Connection Between Linear Algebra and Vector Analysis 1. The Jacobian Matrix For $\mathbff: \mathbbR^n \to \mathbbR^m$, the Jacobian $J$ contains all first partial derivatives: Work done by a force field
Measures flux through a surface. These generalize the Fundamental Theorem of Calculus to higher dimensions: dx + Q
$|\mathbfv| = \sqrt\mathbfv \cdot \mathbfv$
