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A spherical surface F is given by the parametric representation sin(u) cos(v) +1 s(u,v) = | sin(u) sin(v) +1|, u € [0, 7t], v € [0,27] . cos(u) +1 At time t = 0, F starts floating with the vector field. The thereby deformed sur- face is at any time t denoted Ft . Determine a parametric representation of F; and plot F together with F1. %3D

A spherical surface F is given by the parametric representation sin(u) cos(v) +1 s(u,v) = | sin(u) sin(v) +1|, u € [0, 7t], v € [0,27] . cos(u) +1 At time t = 0, F starts floating with the vector field. The thereby deformed sur- face is at any time t denoted Ft . Determine a parametric representation of F; and plot F together with F1. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The flow curve r of a vector field V in (x, y,z) space, which for a given point (xo, yo, zo) fulfils r(0) = (xo, yo, zo) , is given by e' (xo cos(t) – zo sin(t))¯ r(t) = Yoe-t e' (xo sin(t) + zo cos(t)) HOMEWORK SET 7 3 a) A spherical surface F is given by the parametric representation sin(u) cos(v) +1 sin(u) sin(v) + 1 cos(u) +1 s(и, ) — u e [0, 7t] , v E [0, 27t] . At time t = 0, F starts floating with the vector field. The thereby deformed sur- face is at any time t denoted F; . Determine a parametric representation of F; and plot F together with F. b) A solid sphere K with radius a and centre (c1,c2, c3) is given by the parametric representation w sin(u) cos(v)+c1 w sin(u) sin(v) + c2 w cos(u) + сз r(u,v, w) = u € [0, 7] , v € [0,27] , w E [0,a] . At time t = 0, K starts floating with the vector field. The thereby deformed solid is at any time t denoted K; . Does the volume of K; depend on the values of c1, c2 and c3 ? Compute the divergence of V.
Transcribed Image Text:The flow curve r of a vector field V in (x, y,z) space, which for a given point (xo, yo, zo) fulfils r(0) = (xo, yo, zo) , is given by e' (xo cos(t) – zo sin(t))¯ r(t) = Yoe-t e' (xo sin(t) + zo cos(t)) HOMEWORK SET 7 3 a) A spherical surface F is given by the parametric representation sin(u) cos(v) +1 sin(u) sin(v) + 1 cos(u) +1 s(и, ) — u e [0, 7t] , v E [0, 27t] . At time t = 0, F starts floating with the vector field. The thereby deformed sur- face is at any time t denoted F; . Determine a parametric representation of F; and plot F together with F. b) A solid sphere K with radius a and centre (c1,c2, c3) is given by the parametric representation w sin(u) cos(v)+c1 w sin(u) sin(v) + c2 w cos(u) + сз r(u,v, w) = u € [0, 7] , v € [0,27] , w E [0,a] . At time t = 0, K starts floating with the vector field. The thereby deformed solid is at any time t denoted K; . Does the volume of K; depend on the values of c1, c2 and c3 ? Compute the divergence of V.
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