(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are...
(1 point) (a) Show that each of the vector fields F = 4yi + 4xj, G= x y zit vol y J, and ] = vertinant virtuaj are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For F, a potential function is f(x, y) = For G, a potential function is g(x, y) = For i, a potential function is h(x, y) = (b) Find the line integrals of F,...
(1 point) (a) Show that each of the vector fields F-4yi + 4x j, G-i ЗУ x2+y2 x?+yi J, and j are gradient vector fields on some domain (not necessarily the whole plane) x2+y2 by finding a potential function for each. For F, a potential function is f(x, y) - For G, a potential function is g(x, y) - For H, a potential function is h(x, y) (b) Find the line integrals of F, G, H around the curve C...
a) A vector field F is called incompressible if div F = 0. Show that a vector field of the form F = <f(y,z),g(x,z),h(x,y)> is incompressible. b) Suppose that S is a closed surface (a boundary of a solid in three dimensional space) and that F is an incompressible vector field. Show that the flux of F through S is 0. c)Show that if f and g are defined on R3 and C is a closed curve in R3 then...
(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F 1/r2.) The domain of F is R3\{(0, 0, 0)} where r = the chain rule (a) Verify that Hint: first show that then use (b) Show that div(F 0. (c) Suppose that S is a closed surface in R3 that does not enclose the origin. Show that the flux of F through S is zero. Hint: since the interior of S does not contain...
(x2 + y2 + z?)1/2, and e, = r1(x, y, z) is the unit radial vector. Let F = r"e, where n is any number, r= (a) Calculate div(F). (2+n)"-1 (b) Calculate the flux of F through the surface of a sphere of radius R centered at the origin. 4TR"+2 F. ds, where C is a closed curve that does not pass through the origin? (c) What is the value of (d) A function o satisfying Ap = 0 is...
(10) Consider the vector field F (x, y, z) = (x,y, z). Clearly sketch and label three oriented surfaces S, So and S whose flux is negative, zero, and positive, respectively. Be sure to indicate orientations. Explain your conclusions (10) Consider the vector field F (x, y, z) = (x,y, z). Clearly sketch and label three oriented surfaces S, So and S whose flux is negative, zero, and positive, respectively. Be sure to indicate orientations. Explain your conclusions
ems (1 point) A) Consider the vector field F(x, y, z) = (6yz, -7zz, zy). Find the divergence and curl of F. div(F) = V.F= curl(F) = V F =( ). 5 (5x?, 2(x + y), -7(x + y + x)) 7 B) Consider the vector field F(x, y, z) Find the divergence and curl of F. div(F) = V.P= curl(F) = V XF =( 8 9 10 )
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z) = (2x, z, y) + (x, y, z) = (3y, 2x, z). (a) For each field, either find a scalar potential function or prove that none exists. (b) For each field, either find a vector potential function or prove that none exists. (c) Let F(t) = (2, 2t, t2). For which of these vector fields is ñ a flow line? Justify your answer.
DETAILS 3. [2/4 Points) Consider the given vector field. F(x, y, z) = (e", ely, exy?) (a) Find the curl of the vector field. - yzelyz lazenz curl Fe (b) Find the divergence of the vector field. div F = ertxely tuxely F. dr This question has several pa You will use Stokes' Theorem to rewrite the integral and C is the boundary of the plane 5x+3y +z = 1 in the fir F-(1,2-2, 2-3v7) oriented counterclockwise as viewed from...
1. (1.5 points) Sketch the following vector fields: (B) B(x,y)=(z-y,2). (C) Vf where f(x,y) = xy 1. (1.5 points) Sketch the following vector fields: (B) B(x,y)=(z-y,2). (C) Vf where f(x,y) = xy