some formulas

some formulas - Some formulas you may need: Differential :...

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Unformatted text preview: Some formulas you may need: Differential : Given z = f (x, y ), dz = fx dx + fy dy. Given w = f (x, y, z ), dw = fx dx + fy dy + fz dz. Curl : given a vector field F = P, Q, R , curl F = ∂ ∂ ∂ ∂x , ∂y , ∂z × F with = . Jacobian : given the transformation x = x(u, v ), y = y (u, v ), its Jacobian is ∂ (x, y ) = ∂ (u, v ) xu yu xv yv . 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) = ρ2 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, f (r(u, v ))|N(u, v )|dudv, f dS = S D where N(u, v ) = ru (u, v ) × rv (u, v ). Evaluation of the flux integral : Given a surface S : r(u, v ), (u, v ) ∈ D, with n(u, v ) = N(u, v )/|N(u, v )| and a vector field F(x, y, z ) on S, F · dS = S F · ndS = S F(r(u, v )) · N(u, v )dudv, D where N(u, v ) = ru (u, v ) × rv (u, v ). 1 ...
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This note was uploaded on 12/16/2011 for the course MATH 226 at USC.

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