5.) Show that for points z and y on a sphere S there exists a great...
3. If Z and W are two distínct points on the Riemann sphere, then the plane through these points and the origin cuts the sphere in a "great circle," that is, a circle with maximum diameter (2, for a unit sphere). Show that this great circle corresponds to the unique circle (or line) in the plane that passes through the points z, w, and 1/z, where Z, W are the projections of z, w respectively. [HINT: See Prob. 2.] 3....
(1 pt) Consider the sphere (x - 4)2 + (y - 3)2 + (z - 5)2 25 (a) Does the sphere intersect each of the following planes at zero points, at one point, at two points, in a line, or in a circle? The sphere intersects the xy-plane ? The sphere intersects the xz-plane ? The sphere intersects the yz-plane ? (b) Does the sphere intersect each of the following coordinate axes at zero points, at one point, at two...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the boundary of S with counterclockwise orientation when looking down from the z-axis. Verify Stokes' Theorem as follows. (a) (i) Sketch the surface S and the curve C. Indicate the orientation of C (ii) Use the...
Let F(x, y, z) = (yza, x, xy +z) and answer the following questions. Show all work for each part. Q4.3 5 Points Let the surface Si be the part of the unit sphere which sits above the xy-plane. Use Stokes' Theorem to find SSs, curl(F).dS. Please select file(s) Select file(s)
Use Stokes' Theorem to evaluate. 8. Use Stokes, Theorem to evaluate J, ▽ x ที่ do, where F(x, y, z)-(z2yz,yz2,23ezy and s is part of the sphere x2 + y2 + z-5 that lies above the plane z-1. Also, s is oriented upward. 8. Use Stokes, Theorem to evaluate J, ▽ x ที่ do, where F(x, y, z)-(z2yz,yz2,23ezy and s is part of the sphere x2 + y2 + z-5 that lies above the plane z-1. Also, s is oriented...
Suppose that A, B, C are three non-collinear points in a plane. Show that there exists a circle that passes through all 3. ($16.4, # 2)
(Complex analysis) Exercise 6 a) Show that the image of the half-plane y > c (c = const) in the z-plane 1 under the inversion mapping w--s the interior of a circle provided that C0 the inversion mapping w hen0? the inversion mapping w = z when c < 0? b) What is the image of the half-plane y > c (c -const) in the z-plane under c) What is the image of the half-plane y > c (cconst) in...
1. Suppose F = (-y,x,z) and S is the part of the sphere x2 + y2 + z = 25 below the plane z = 4, oriented with the outward-pointing normal (so that the normal at (5,0,0) is 1). Compute the flux integral curl F.ds using Stoke's theorem.
(5) Circle each abbreviation, O-open, plies to the given subset of S cC: C-closed, D domain, CP compact, which ap CD CP S,^ i....) (6) a) Use the formulas for chordal distance to show that χ[1/리, 1/22-11, 22], for any 21,22 E C. In other words, the reciprocal mapping R(z) l/s preserves the chordal distance between two points. (Here the definitions 1 /0 = oo. 1/00-0 apply.) b) Use the formula for inverse stereographic projection (Equation 1.7.1) to show that...