MAT362Homework5

MAT362Homework5 - MAT 362 HOMEWORK 5 . Exercise 27 ^ grad =...

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Unformatted text preview: MAT 362 HOMEWORK 5 . Exercise 27 ^ grad = 1 r 4 grad r = 4r3 r ^ div r = 1 ^ r curl grad = 0 div grad = 2 = 0 ^ div curl = 0 . Exercise 29 x J= (x, y, z) = (r, , ) r y r z r x y z x y z cos() sin() -r sin() sin() r cos() cos() = sin() sin() r cos() sin() r sin() cos() cos() 0 -r sin() a= b= c= cos2 () sin2 () + sin2 () sin2 () + cos2 () = 1 r2 sin2 () sin2 () + r2 cos2 () sin2 () = r sin() r2 cos2 () cos2 () + r2 sin2 () cos2 () + r2 sin2 () = r 1 1 grad = - 2 r ^ r r 1^ grad = r curl grad = 0 div grad = 2 = r12 cot() div grad r = 2 r = 6 because r = x2 + y 2 + z 2 in cartesian coordinates, and therefore, div grad r = 2 + 2 + 2 = 6 . Exercise 30 We have u = r cos() + so grad u = (cos() - 1 cos(), r2 1 1 ^ cos())^ + (- sin() - 3 sin()), r r3 2r 2 1 2 1 div grad u = cos() + 4 cos() - cos() - 4 cos() = 0. r r r r Thus, div grad u = 2 u = 0. 1 ...
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