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some formulas

# some formulas - Some formulas you may need Dierential Given...

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Some formulas you may need: Differential : Given z = f ( x, y ), dz = f x dx + f y dy. Given w = f ( x, y, z ) , dw = f x dx + f y dy + f z dz. Curl : given a vector field F = h P, Q, R i , curl F = ∇ × F with = D ∂x , ∂y , ∂z E . Jacobian : given the transformation x = x ( u, v ) , y = y ( u, v ), its Jacobian is ( x, y ) ( u, v ) = x u x v y u y v . Spherical coordinates : x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ ; absolute value of the Jacobian: ( x,y,z ) ( ρ,θ,φ ) = ρ 2 sin φ. Evaluation of the surface integral : Given a surface S : r ( u, v ) , ( u, v ) D, and a function f ( x, y, z ) on S, Z Z S fdS = Z Z D f ( r ( u, v ))
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