Problem 1.20. Let f(z, y)-(X2-y2)/(z2 + y2) 2 for x, y E (0, 1]. Prove that...
Let F(x, y,z) = < x + y2,y + z2,z + x2 >, let S be a surface with boundary C. C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1). 8. a. Evaluate F dr curl F ds b. Let F(x, y,z) = , let S be a surface with boundary C. C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1). 8. a. Evaluate F dr curl F ds b.
The gravitational field F(x,y,z) =cx /(x2 + y2 + z2)3/2 e1+ cy /(x2 + y2 + z2)3/2 e2+ cz/ (x2 + y2 + z2)3/2 e3 is a gradient field, where c is a constant, such that the field is rotation free. If we define f(x,y,z) = −c /(x2 + y2 + z2)1/2 , then show that (a) F = grad(f). (b) curl(F) = 0.
Let C1 be the semicircle given by z = 0,y ≥ 0,x2 + y2 = 1 and C2 the semicircle given by y = 0,z ≥ 0,x2 +z2 = 1. Let C be the closed curve formed by C1 and C2. Let F = hy + 2y2,2x + 4xy + 6z2,3x + eyi. a) Draw the curve C. Choose an orientation of C and mark it clearly on the picture. b) Use Stokes’s theorem to compute the line integral ZC...
Evaluatef(x, y, z) dS. f(x, y, z) = x2 + y2 +z2 Evaluatef(x, y, z) dS. f(x, y, z) = x2 + y2 +z2
Problem 1: Consider the following problem x+y+1=1 x2 +y2+z2 =1 max f(x ,y,z)=er+y+1 subject to (a) Solve the problem. (b) Replace the constraints byx+y+1=1.02 and x2+y2+Z2-0.98. What is the approximate change in the optimal value of the objective function? (c) Classify the candidate points for optimality in the local optimization problem.
evaluate JJ. (< –Y) A. ) Integrate f(x, y, z) = x2 + y2 + 22 over the cylinder x2 + y2 < 2,-2 <2<3 (IL dx dy dz Feraluate
10. Let F(x, y, z) = 〈y,-z, 10) per half of x2 +y2 + z2 = 1, oriented upward, and C the circle 2 y 1 in the z - y plane, oriented counter-clockwise. Find Jscurl(F) ndS directly and by using Stokes' Theorem. , where S is the up 10. Let F(x, y, z) = 〈y,-z, 10) per half of x2 +y2 + z2 = 1, oriented upward, and C the circle 2 y 1 in the z - y...
please help me solve the following question 8. Compute JJ f dS where f(x, y, 2)22+2 and S is the top hemisphere x2 + y2 + Z2, 220. 9. Compute JJ F-n dS where F-: (x, y, z) and s is the cone z2 x2 + y2, 0 S 2 1; with the outward pointing normal. 8. Compute JJ f dS where f(x, y, 2)22+2 and S is the top hemisphere x2 + y2 + Z2, 220. 9. Compute JJ...
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that: 1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...