Course Note
This lesson extends the algebraic toolkit you have built — manipulating expressions, solving equations, working with functions — into the territory of multivariable calculus. The theorems below are the capstone of vector analysis, but every step rests on the same algebraic discipline: substitution, factoring, careful sign-tracking, and systematic evaluation of integrals. Think of this as algebra at its most powerful.
1. The Architecture: One Theorem Wearing Dimensions
Before we state anything, let us see the pattern that runs through every major theorem in calculus:
This pattern first appears in the Fundamental Theorem of Calculus (Newton–Leibniz):
$$ \int_a^b f'(x)\,dx \;=\; f(b) - f(a) $$The derivative $f'$ is integrated over the interval $[a,b]$; the result is $f$ evaluated at the two boundary points.
Green's Theorem is the plane case — a 2D region $D$ with boundary curve $\partial D$:
$$ \oint_{\partial D} \mathbf{F}\cdot d\mathbf{r} \;=\; \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA $$Now the two theorems of this lesson complete the family:
- Stokes' Theorem — surface $S$ with boundary curve $\partial S$ (Green's theorem lifted into 3D).
- Divergence Theorem — solid $V$ with boundary surface $\partial V$ (Green's normal-form lifted into 3D).
All four are special cases of the generalized Stokes theorem on differential forms:
$$ \int_M d\omega \;=\; \int_{\partial M} \omega $$