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# ChapterHigher Dimensions and Other J16 Coordinatet Because differentiation with respect to X,y or gives the with respect t0 2, and because = samne outside the integ...

## Question

###### ChapterHigher Dimensions and Other J16 Coordinatet Because differentiation with respect to X,y or gives the with respect t0 2, and because = samne outside the integral the boundary 5 resekt inwt , (12), we find that UTNRY $dz = 02v ax?and satisfies the two-dimensional heat equation azv 8-v Ju 0 < x < a, Dy2 k at 0 < Y < b, 0 < [ (See the exercises for details and for boundary and initial conditions } If z-variation cannot be ignored, we could try to get rid of by introducing an Chapter Higher Dimensions and Other J16 Coordinatet Because differentiation with respect to X,y or gives the with respect t0 2, and because = samne outside the integral the boundary 5 resekt inwt , (12), we find that UTNRY$ dz = 02v ax? and satisfies the two-dimensional heat equation azv 8-v Ju 0 < x < a, Dy2 k at 0 < Y < b, 0 < [ (See the exercises for details and for boundary and initial conditions } If z-variation cannot be ignored, we could try to get rid of by introducing an average in that direction, the Yavariaua w(x, 20=b J ulx, Y,2,t)dy. From the boundary condition (13) we find that du Ju 8yzdy (x,b,2,t) _ (x, 0,2,t) [(Tz u(x, b,2,t)) + (Tz ulx, '0,2,t)11 . If b is small-_-the parallelpiped is more like plate_we approximation u(x_ may b.2,t) + u(x, 0, 2,t) = Zw(x, 2,t) , which accte" t from the expression above, would Iak: Ody = 2h b bk (Tz w(x,2,t)) _ After applying the averaging process to Eqs: (10), (11), (12) and (I4) weobzz the following two- ~dimensional problem for W. 02w 92w 2h dw axl bk (T-w) = k au 0 < x < a, 0 < 2 < â‚¬, 0 <t / w(0,2,t) = To, w(a,2,t) = T;, 8w 0 < 2 < â‚¬, 0 < t (x,0,t) = 0, dw d2 dz (x,c,t) = 0, 0 < x < a, 0 < t w(x,2,0) = 6 k" f6xsy,z)dy, 0 < x < a, 0 < 2 < â‚¬  #### Similar Solved Questions

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