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finalformulas - Formula sheet for final exam in calc IV,...

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Unformatted text preview: Formula sheet for final exam in calc IV, summer 2007 Polar coordinates x = r cos θ y = r sin θ r2 = x2 + y 2 dA = rdrdθ Cylindrical coordinates x = r cos θ y = r sin θ r2 = x2 + y 2 dV = rdzdrdθ Spherical coordinates x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ ρ2 = x2 + y 2 + z 2 dV = ρ2 sin φdρdθdφ Change of variables in 2 dimensions f (x, y ) dAx,y = R (f ◦ T )(u, v )|J (T )| dAu,v ; J (T ) = S ∂ (x,y ) ∂ (u,v ) = ∂x ∂u ∂y ∂u ∂x ∂v ∂y ∂v , the Jacobian. Change of variables in 3 dimensions R f (x, y, z ) dVx,y,z = S (f ◦ T )(u, v, w)|J (T )| dVu,v,w ; J (T ) = ∂ (x,y,z ) ∂ (u,v,w) = ∂x ∂u ∂y ∂u ∂z ∂u ∂x ∂v ∂y ∂v ∂z ∂v ∂x ∂w ∂y ∂w ∂z ∂w Line Integrals C f (x, y )ds = b a dx 2 dt f ((x(t), y (t)) + dy 2 dt dt b a F · dr = F(r(t)) · r (t) dt b P (x, y ) dx + Q(x, y )dy = a P (x(t), y (t))x (t) dt + Q(x(t), y (t))y (t) dt C C Green’s Theorem C P dx + Q dy = R ∂Q ∂x − ∂P ∂y dA Surface/Flux Integrals f (x, y, z ) dS = D f (r(u, v ))|ru × rv | dAuv = D f (r(u, v )) |ru |2 |rv |2 − (ru · rv )2 dAuv S ˆ F · dS = S F · n dS = D F(r(u, v )) · (ru × rv ) dAuv S ∂r ∂r Parametrically: r(u, v ) = x(u, v )i + y (u, v )j + z (u, v )k, ∂u × ∂v is a normal to S , dS = ∂r ∂r | ∂u × ∂v | dAuv , |ru × rv | = |ru |2 |rv |2 − (ru · rv )2 so if the parametrization is orthogonal (ie ru · rv = 0) then dS = |ru × rv | = |ru ||rv |. As graph: z = f (x, y ), − ∂f i − ∂x ∂f j ∂y + k is a normal to S , dS = ∂ ∂ ∂ = i ∂x + j ∂y + k ∂z Stokes’ Theorem ˆ ( × F) · n dS = S ∂S F · dr Divergence Theorem ˆ · F dV = ∂ E F · n dS E 1 ∂f 2 ∂x + ∂f ∂y 2 + 1 dAxy . ...
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