Useful Facts

Useful Facts - Math E-21a Some useful facts f dx f dy + = f...

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Math E-21a – Some useful facts Basic Chain Rule : [] (() , () ) fd x y d f xt yt f dt xd t yd t ∂∂ =+= v for a path in R 2 ; ,() ) x y z d f xt yt zt f dt x dt y dt z dt ∂∂∂ =++= v for a path in R 3 . Directional Derivative of a function f in the direction u (unit vector): Df f =∇⋅ u u Fundamental Theorem of Line Integrals : If V is differentiable and C is a curve from point x 0 to point x 1 , then 10 () () C Vd r V V ∇⋅ = xx !!!" !!" . Green’s Theorem : If (, ) Pxy and Qxy are differentiable with continuous 1 st partial derivatives through a bounded region D in R 2 and if C is the boundary of D oriented in the counterclockwise sense, then CD QP P x y dx Q x y dy dA xy  +=   ∫∫ # . Divergence Theorem : If the components of the vector field F are differentiable with continuous 1 st partial derivatives through a bounded region B in R 3 and if S is the boundary of B oriented with unit outward normal vector n , then () ( ) div SS B dd S d V ⋅= = ∫∫∫ FS Fn F $$ . Curl and Divergence : If (, ,) , (, ,) xyz Pxyz Qxyz Rxyz = F is a vector field in R 3 with differentiable component functions, then curl , , R QP RQ P yz zx = −−− F and div PQR x =++ 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.

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