Vector Analysis question B.) (2pts) Let f: ECR” → R be defined as where x =...
Problem #7: Let R = r \ {(0,0,0)) and F is a vector field defined on R satisfying curl(F) = 0. Which of the following statements are correct? [2 marks] (1) All vector fields on R are conservative. (ii) All vector fields on Rare not conservative. (iii) There exists a differentiable function / such that F - Vf. (iv) The line integral of Falong any path which goes from (1,1,1) to (-2,3,-5) and does not pass through the origin, yields...
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...
(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...
(b) Let F: R2 + Rº be a vector field on R2 defined as F(x, y) = (3y, 22 – y). Suppose further that ^ C R2 is a curve in R2 consisting of the parabola y = 22 - 1 for 1 € (-1,0) and the straight line y = 1 – 1 for 1 € [0,1]. (i) Sketch the curvey in R2 [2] (ii) By considering the curve y piecewise, compute the vector field integral: [5] F(x). F(x)...
Analysis problem
(b) Let f, q be defined on A to R and let c be a cluster point of A i. Show that if both lim f and lim (f + g) exist, then lim g exists. c I>c ii. If lim f and lim fg exist, does it follow that lim g exists? -c (c) Suppose that f and g have limits in R as x -> o and that f(x) < g(x) for all x € (a,...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
Let F = (P,Q) be the vector field defined by -x+y . P(x,y) = 22, (x, y) + (0,0) 0, (x, y) = (0,0) Q(x,y) = (x, y) + (0,0) x2+y2; 10,(x, y) = (0,0). (a) Show that F is a gradient vector field in R2 \ {y = 0}. (b) Letting D = {2:2020 + y2020 < 1}, compute the line integral Sap P dx + Q dy in the counter-clockwise direction. (c) Does your calculation in part (b)...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
21 Problem 20. Let S be the surface bounded by the graph of f(x,y)-2+y2 . the plane z 5; Os1; and .0sys1. In addition, let F be the vector field defined by F(x, y,z):i+ k. (1) By converting the resulting triple integral into cylindrical coordinates, find the exact value of the flux integral F.n do, assuming that S is oriented in the positive z-direction. (Recall that since the surface is oriented upwardly, you should use the vector -fx, -fy, 1)...
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(a) Let v(r) be a scalar function of r, where r V +y? +22 (i) Show that (i) If F Vu) evaluate Jc Fdr where C is straight line going from the point defined by vector r1 to the point defined by r2 (b) Consider a body with a surface defined by 2(x2 + y2) + 4z2 1 (i)...