Dr. Z’s Math251 Handout #16.9 [The Divergence Theorem]By Doron ZeilbergerProblem Type 16.9a: Use the Divergence Theorem to calculate the surface integralR RSF · d S,whereF(x, y, z) = P (x, y, z) i + Q(x, y, z) j + R(x, y, z)k ,where S is a surface bounding some solid region that can be expressed in some coordinate system(Cartesian, cylindrical, or spherical).Example Problem 16.9a: Use the Divergence Theorem to calculate the surface integralR RSF · d S, whereF(x, y, z) = e2xsin 2y i + e2xcos 2y j + y2z2k ,where S is the surface of the box bounded by the planes x = 0, x = 2, y = 0, y = 1, z = 0, z = 3.Steps Example1. Compute the divergence of F:div F =∂P∂x+∂Q∂y+∂R∂z.1.div F =∂∂x(e2xsin 2y)+∂∂y(e2xcos 2y)+∂∂z(y2z2)= 2e2xsin 2y−2e2xsin 2y+2y2z = 2y2z .2. Write the solid region E in ‘iteratedform’ in either Cartesian, Cylindrical, orSpherical coordinates.2. The solid region E bounding our sur-face is clearly the box{ (x, y, z) | 0 ≤ x ≤ 2 , 0 ≤ y ≤ 1 , 0 ≤ z ≤ 3 } .13. Set-up the Divergence Theorem, andevaluate the triple integral.Z ZSF · d S =Z Z ZEdiv F dV .3.Z ZSF · d S =Z Z ZEdiv F dV .=Z20Z10Z302y2z dz dy dx .= 2Z20dxZ10y2dyZ30z dz= 2(2−0)·13− 033·32− 022= 2·3 = 6 .Ans.:
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