help pleasee
ignore the first photo, i need help with number
2
help pleasee ignore the first photo, i need help with number 2 4. Let F(x, y,...
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
8Two vector fields are given: F(x,y,z) - (esin(yz), ze* cos(yz), ye* cos(yz)) and F(x,y,z) = (z cos y, xz sin y, x cos y). a) Determine which vector field above is conservative. Justify. Foly = fjol so, <ea sin(J2), 20% cos(82), y acos (92)) Conservative. b) For the vector field that is conservative, find a function f such that F - Vf. Lxelsing2, zetos yea, yet cosy 2 c) Use the Fundamental Theorem of Line Integrals to find the work...
Show all work and use correct notation for full credit. Stokes' Theorem: Let S be an orientable, piecewise smooth surface, bounded by a simple closed piecewise smooth curve C with positive orientation. Let F be a vector field with component functions that have continuous partial derivatives on an open region in R3 that contains S. Then | | curl(F) . ds F-dr = where curl(F) = ▽ × F. (2 Credits) Let S be the cone given by {(z, y,...
3. Let F(x, y) = (-y, x) and let C be the semicircle with parametrization f(t) = (4 cos t, 4 sin t) for 0 <ts. Compute ScF.T ds.
(6) Show that F(x, y) = (x+y)i + (**)is conservative. (a) Then find such that S = F (potential function). (5) Use the results in part(a) to cakulae ( F. ds along C which the curve y = a* from (0,0) to (2,16). (2) Use Green's Theorem to evaluate 1. F. ds. F(1,y) =(yº+sin(26))i + (2xy2 + cos y)and C is the unit circle oriented counter clockwise (6) Evaluate the surface integral || 9. ds. F(x,y,z) = xi +++where S...
you can skip #2 Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u where f(r,y,) = =- +22 2. Consider the vector field F(E,) = (a,y) Compute the flow lines for this vector field. 3. Compute the divergence and curl of the following vector field: F(x,y,)(+ yz, ryz, ry + 2) Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u...
Thank you to any who can help. - Let f(x, y) = sin r cos y +2.14 and let C be the path in the ry plane that follows the arc of y = sin x from 6, 1) to (7,0). Then (a) Find the gradient of f, that is, find F=Vf. (b) Explain why Green's Theorem cannot be applied to find | F. dr Jc (c) Use a different method to find F. dr
1. (2 points) Find F dF if curl(F) 3 in the region defined by the 4 curves and C4 Ci F . d7 where F(x,y,z)-Wi +pz? + Vi> and C consists of the arc of the 2. (2 points) Evaluate curve y = sin(x) from (0,0) to (π, 0) and the line segment from (π,0) to (0,0). 4 3 3. (2 points) Evaluate F di where F.y,(ry, 2:,3) and C is the curve of intersection of 5 and y29. going...
Let y: 1 + R2 be a regular parametrised curve which we write as y(t) = (v(t), v(t))" for some smooth maps u,v: 1 R. We assume furthermore that is never equal to zero on I. We define the surface of revolution Exy associated to y as (1) E = {r(t,0) = (v(t) cos(6), y(t) sin(0), v(0))?|tel, 0 € (0,27]} . Below, we consider the chart (U,r) obtained by taking U = I x (0,27), where the map r:U →...
Question 12 11. Show that if F is continuous on Rn and F(X + Y) = F(X) + F(Y) for all X : in R", then A is linear. HINT: The rational numbers are dense in the reals. 12. Find F and JF. Then find an affine transformation G such that F(X)-G(Y) lim =0. T x2+y+2z (a) F(x, y,z)coxy. Xo- (1,-1,0) e*yz ex cos y (b) Fe*sin y 1, xo=(0, π/2) 13. Find F. g1 (x) 11. Show that if...