Math 192 Cheat Sheet - d d d dV z r z r y r x r sin , cos ,...

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Unformatted text preview: d d d dV z r z r y r x r sin , cos , sin , cos , sin 2 2 2 = + = = = = = Line Integral ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 = + + = b a C dt t t k t h t g f ds z y x f b t a t k t h t g t v , , , , k, j i r Mass ( ds = d for thin shells) ( 29 M M z M M y M M x ds z M ds y M ds x M ds z y x M xy xz yz C xy C xz C yz C = = = = = = = , , , , , , Work + + = + + = = = = = b a b a b a b t a t dz P dy N dx M dt dt dz P dt dy N dt dx M dt d ds W dt r F T F Work done by conservative field ( 29 - = = B A B A A f B f d f d ) ( r r F Flow = = = b t a t dt d dt r F Flow Flux Across Closed Curve in the Plane j i k j i k T n n F across F of Flux ds dx ds dy ds dy ds dx ds C C- = + = = = Flux Across a Smooth Closed Plane Curve (traces counterclockwise) ( 29 - = + = C Ndx Mdy C N M across j F of Flux i Test for Conservative Field or Exactness of M dx + N dy + P dz y M x N x P z M z N y P = = = and , , EX: Finding Potential Function ). ( ) ( to equal is it integral, For C 2 cos ) , , ( C 2 ) ( so or , ) ( cos ) , , ( so , sin sin ) , ( cos ) , , ( , sin , cos exact or ve conservati is it if Test ) ( ) sin ( ) cos ( or k ) ( j ) sin ( i ) cos ( F 2 2 A f B f z xyz y e z y x f z z h z dz dh z xy dz dh xy z h xyz y e z y x f y g y e xz y g xz y e z y g xyz y e z y x f z xy z f y e xz y f yz y e x f dz z xy dy y e xz dx yz y e z xy y e xz yz y e x x x x x x x B A x x x x- + + + = + = = + = + + + = = - = + +- + + = + = - = + = + +- + + + +- + + = Divergence (Flux Density) of F = M i + N j+ P k z P y N x M + + = F div Circulation Density of F =...
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This note was uploaded on 02/28/2008 for the course MATH 1920 taught by Professor Pantano during the Fall '06 term at Cornell University (Engineering School).

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Math 192 Cheat Sheet - d d d dV z r z r y r x r sin , cos ,...

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