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Summary of Theorems

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    39458
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    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

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


    This page titled Summary of Theorems is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

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