(a) The relationships between polar coordinates in the plane and rectangular coordinates are given by x...

  

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(a) The relationships between polar coordinates in the plane and rectangular coordinates are given by x = r cos 0, y = r sin 0, r = x + y and tan 0 = y/x Determine functions A(0), B(0), C(r,0) and D(r, 8) such that u(x,y) can be expressed as = A(0) + C(r,0) ar == B(0)- +D(r,0) r (b) Determine functions R(r) and S(r) such that the two-dimensional Laplacian of u(x,y) can further be written as vu = + = +R(r). +S(r) 2 r2 r 02 (c) Determine the groundwater depth h(r, 0) which satisfies the Laplace's equation for the two-dimensional steady groundwater flow and the boundary conditions. ah 1h 1 Jh + ar2 rar 2002 0 ah h h(a,0)=h, -(b,0)= 00 <2, 0 (a) The relationships between polar coordinates in the plane and rectangular coordinates are given by x = r cos 0, y = r sin 0, r = x + y and tan 0 = y/x Determine functions A(0), B(0), C(r,0) and D(r, 8) such that u(x,y) can be expressed as = A(0) + C(r,0) ar == B(0)- +D(r,0) r (b) Determine functions R(r) and S(r) such that the two-dimensional Laplacian of u(x,y) can further be written as vu = + = +R(r). +S(r) 2 r2 r 02 (c) Determine the groundwater depth h(r, 0) which satisfies the Laplace's equation for the two-dimensional steady groundwater flow and the boundary conditions. ah 1h 1 Jh + ar2 rar 2002 0 ah h h(a,0)=h, -(b,0)= 00 <2, 0