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 counterclockwise 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

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