Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
16, Let x: U R2-, R, where x(8, φ) (sin θ cos φ, sin θ sin φ, cos θ), be a parametrization of the unit sphere S2. Let and show that a new parametrization of the coordinate neighborhood x(U) = V can be given by y(u, (sech u cos e, sech u sin e, tanh u Prove that in the parametrization y the coefficients of the first fundamental form are Thus, y-1: V : S2 → R2 is a conformal...
6.8. Verify that u(x, y)= A sin(27Tx) sin(27ty) solves Poisson's equation V2u-Ron W (0, 1) x (0,1) for some A-value, where R(x, y) sin(2x) sin(2Ty) (a) Find the correct A value (b) Compute the total source S w RdA (c) Compute the flux out through the top part of W (y 1) and verify by symmetry that it is one-quarter that of the full source S. 6.8. Verify that u(x, y)= A sin(27Tx) sin(27ty) solves Poisson's equation V2u-Ron W (0,...
a. Use the Chain Rule to find the indicated partial derivatives. z = x4 + x2y, x = s + 2t − u, y = stu2; ∂z ∂s ∂z ∂t ∂z ∂u when s = 1, t = 2, u = 3 b. Use the Chain Rule to find the indicated partial derivatives. w = xy + yz + zx, x = r cos(θ), y = r sin(θ), z = rθ; ∂w ∂r ∂w ∂θ when r = 8, θ = pi/2 c. Use the...
Give a parametric description of the form r(u, v) = (x(u,v),y(u,v),z(u,v)) for the following surface. The cap of the sphere x² + y2 + z = 36, for 3/3 sz56 Select the correct choice below and fill in the answer boxes to complete your choice. (Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than 21. Type exact answers in terms of ..) A. Fu.v) = (6 sin u cos V,6 sin...
V X2 + y2 and θ u(r(z, y), θ(x, y))--sech2 r tanh r sin θ 6. [Sec. I 1.5] Letr tan l (y/z) be the usual polar rectangular coordinates relationships. Furthermore, define and u(r(z, y),θ(z, y)) sech2 r tanh r cos θ Show that tanh r
The Laplacian and harmonic functions The quantity V-Vu-V2u, called the Laplacian of the function u, is particularly useful in applications. (a) For a function u(x, y, z), compute V Vu (c) A scalar valued function u is harmonic on a region D if V a all points of D. Compare this to Laplace's equation eu +Pn=0 and ψ" + ψ”=0. The Laplacian and harmonic functions The quantity V-Vu-V2u, called the Laplacian of the function u, is particularly useful in applications....
4.let U= {q,r,s,t,u,v,w,x,y,z}; A= {q,s,u,w,y};and C={v,w,x,y,z,}; list the members of the indicated set , using set braces A'u B A.{Q,R,S,T,V,X,Y,Z} B.{S,U,W} C.{R,S,T,U,V,W,X,Z} D.{Q,S,T,U,V,W,X,Y}
7. Show that the following functions u(x, y) monic functions v(x, y) and determine f(z) = u(x,y) + iv(x, y) are harmonic, find their conjugate har- as functions of 2. 2x2 2лу — 5х — 22. Зл? — 8ху — Зу? + 2у, (а) и(х, у) (b) и(х, у) (с) и(х, у) (d) u(a, y) 2e cos y 3e" sin y, = 3e-* cos y + 5e-" sin y, = elx cos y - e y sin y, (e) u(x,...
Express the unit vectors, , φ in terms of 8,9, 2 (that is, derive Eq. 1.64). Check your answers several ways ( f . f = 1,0 . φ 0, f × ยิ p, you need to check all of these explicitly, the solution manual does not give all of these). Also work out the inverse formulas, giving £,9,2 in terms of f, θ and θ, φ). sn θ cos φ x + sin θ sin φ y + cos...