Summary of Theorems

Fundamental Theorem of Line Integrals

Suppose a curve $C$ is given by the vector function ${\bf r}(t)$, with ${\bf a}={\bf r}(a)$ and ${\bf b}={\bf r}(b)$. Then

$$\int_C \nabla f\cdot d{\bf r} = f({\bf b})-f({\bf a}),$$

provided that $\bf r$ is sufficiently nice.

Green's Theorem

If the vector field ${\bf F}=\langle P,Q\rangle$ and the region $D$ are sufficiently nice, and if $C$ is the boundary of $D$ ($C$ is a closed curve), then

$$\iint\limits_{D} {\partial Q\over\partial x}-{\partial P\over\partial y} \,dA = \int_C P\,dx +Q\,dy ,$$

provided the integration on the right is done counter-clockwise around $C$.

Stoke's Theorem

Provided that the quantities involved are sufficiently nice, and in particular if $D$ is orientable,

$$\int_{\partial D} {\bf F}\cdot d{\bf r}=\iint_\limits{D}(\nabla\times {\bf F})\cdot{\bf N}\,dS,$$

if $\partial D$ is oriented counter-clockwise relative to $\bf N$.

Green's Theorem( 3D)

If the vector field ${\bf F}=\langle P,Q\rangle$ and the region $D$ are sufficiently nice, and if $C$ is the boundary of $D$ ($C$ is a closed curve), then

$$\int_{\partial D} {\bf F}\cdot{\bf N}\,ds=\iint\limits_{D} \nabla\cdot{\bf F}\,dA. \nonumber$$

Divergence Theorem

Under suitable conditions, if $E$ is a region of three dimensional space and $D$ is its boundary surface, oriented outward, then

$$\iint\limits_{D} {\bf F}\cdot{\bf N}\,dS=\iiint\limits_{E} \nabla\cdot{\bf F}\,dV. \nonumber$$