Curl i curl f f x f1 j y f2 k z f3 if the domain

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Unformatted text preview: er a region R in R2 . Area of R = R dA. f (x, y) dA. R Volume of the solid under the surface z = f (x, y), over R = In polar coordinates: R f (x, y) dA = R f (r cos , r sin )r d dr. Divergence div F = F = F1 F2 F3 + + . x y z 5 Green's theorem F dr = F1 dx + F2 dy = C R C F2 F1 - x y dx dy . In vector form: C F dr = C R ( F) k dA. R In divergence form: Surface integrals F n ds = F dx dy = Flux of F across C. (x, y, z) dS = S R 2 2 (x, y, f (x, y)) 1 + fx + fy dx dy In cylindrical coordinates: In spherical coordinates: f (x, y, z) dS = Flux across S = S (x, y, z) dS = (a cos , a sin , t) a d dt. f (a cos sin , a sin sin , a cos ) a2 sin d d . F n dS. Triple integrals In cylindrical coordinates: (x, y, z) dx dy dz = In spherical coordinates: f (x, y, z) dx dy dz = Divergence theorem F n dS = F dV f r cos sin , r sin sin , r cos r 2 sin dr d d r cos , r sin , t r dr d dt S V Stokes' theorem C F dr = S ( F) n dS 6 Table of Standard Integrals 1. xn dx = xn+1 +C n+1 (n = -1) 6. sec2 x dx = tan x + C 2. dx = ln |x| + C x 7. sinh x dx = cosh x + C ex dx = ex + C 8. cosh x dx = sinh x + C dx = sin-1 2 - x2 a 3. 4. sin x dx = - cos x + C 9. cos x dx = sin x + C dx = + x2 x a +C 5. 10. a2 1 a tan-1 x a +C 11. dx = sinh-1 2 + a2 x dx = cosh-1 2 - a2 x x a + C = ln x + x2 + a2 + C 12. x a +C (x > a) (x > a or x < -a) = ln x + x2 - a2 + C THIS IS THE END OF THE QUIZ PAPER. 7...
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