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.
The gravitational field F(x,y,z) =cx /(x2 + y2 + z2)3/2 e1+ cy /(x2 + y2 +...
Evaluatef(x, y, z) dS. f(x, y, z) = x2 + y2 +z2 Evaluatef(x, y, z) dS. f(x, y, z) = x2 + y2 +z2
Consider the given vector field. F(x, y, z) = (9 / sqrt(x2 + y2 + z2)) (x i + y j + z k) Find the curl of the vector field. Then find Divergence
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.
3)If w = x2 + y2 + z2 ; x = cos st, y = sin st , z = sat find 4)Find the minimum of the function f(x,y) = x2 + y2 subject to the constraint g(x, y) = xy - 3 = 0 5)Find the first and second order Taylor polynomials to the function f(x,y) = ex+y at (0,0). 6) Let f(x, y, z) = x2 – 3xy + 2z, find Vf and Curl(f)
Problem 1.20. Let f(z, y)-(X2-y2)/(z2 + y2) 2 for x, y E (0, 1]. Prove that f(x, y) dx dy f f(x,y) dy)dr. Jo Jo JoJo
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...
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...
*PLEASE SHOW ALL WORK AND STEP BY STEP SOLUTION* 7.+-/3 points MarsVectorCalc6 2.6.011. For the functions below, what is the direction of fastest increase at (1, 1, 1)? 8 (a) f(x, y 2) -2+ y2 +2 x2 + y2 + z2 (b) f(x, y, z)-2xy + 2yz + 2xz 8 (cx,2x2 + y2+ 2 2 + z2 7.+-/3 points MarsVectorCalc6 2.6.011. For the functions below, what is the direction of fastest increase at (1, 1, 1)? 8 (a) f(x, y...
Use Lagrange multipliers to find the min and max of f(x,y,z) = x2-y2+ 2z subject to the constraint x2 + y2 + z2 = 1.
6 (20 pts). Let F(x, y, z) = x2 + y2 + x2 - 6xyz. (1) Find the gradient vector of F(x, y, z); (2) Find the tangent plane of the level surface F(x, y, z) = x2 + y2 + x2 - 6xyz = 4 at (0, 0, 2); (3) The level surface F(x, y, z) = 4 defines a function z = f(x,y). Use linear approxi- mation to approximate z = = f(-0.002,0.003).