Math E-21a – Some useful facts Basic Chain Rule: ((), ())fdxydf xt ytfdtxdtydt∂∂=+=∇⋅vfor a path in R2; ,())xyzdf xt yt ztfdtx dty dtz dt∂∂∂=++=∇⋅vfor a path inR3. Directional Derivativeof a function fin the direction u (unit vector): Dff=∇⋅uuFundamental Theorem of Line Integrals: If Vis differentiable and Cis a curve from point x0to point x1, then 10() ()CVdr VV∇⋅ =−∫xx!!!" !!". Green’s Theorem: If (, )Pxyand Qxyare differentiable with continuous 1stpartial derivatives through a bounded region Din R2and if Cis the boundary of Doriented in the counterclockwise sense, then CDQPP x y dxQ x y dydAxy+=−∫∫∫#. Divergence Theorem: If the components of the vector field Fare differentiable with continuous 1stpartial derivatives through a bounded region Bin R3and if Sis the boundary of Boriented with unit outward normal vector n, then ()( )div SSBddSdV⋅=⋅=∫∫∫FSFnF$$. Curl and Divergence: If (, ,), (, ,)xyzPxyz Qxyz Rxyz=Fis a vector field in R3with differentiable component functions, then curl,,RQP RQ Pyzzx=−−−Fand div PQRx=++F. Surface integration “toolkits”:Sphere of radius
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This note was uploaded on 02/10/2011 for the course MATH E-21a taught by Professor - during the Fall '09 term at Harvard.