. Calculate .F for the tollowing vector ficlds: (a) F (r, 2y,3z) (c) F (32,y2,22) (b)...
Calculate ▽ × F for the following vector fields: (a) F (r, 2y,32) (c) F (32,y2,.2), d) F (y,yz, a),
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y...
Solve c and d Please.
Find the area of the following surfaces. a) The part of the plane 3z + 2y +z 6 that lies in the first octant. b) The part of the cone zVy that lies between the plane y z and the cylinder y-x c) The surface with vector equation r(u, u) = (u . cos(v), u . sin(v), u) , where 0 u 1, 0 v S T. d) The portion of the unit sphere that...
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
Use (part A) line integral directly then use (part B) Stokes'
Theorem
10. Use Stokes's Theorem to evaluate F dr where F(x, y, z) (3z 2y)i + (4x 3y)j + (z + 2y)k and C is the unit circle in the plane z (a) 67 (d) 12m 3. (b) TT (e) None of these (c) 3 TT
10. Use Stokes's Theorem to evaluate F dr where F(x, y, z) (3z 2y)i + (4x 3y)j + (z + 2y)k and C...
(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...
For the reaction X(g)+2Y(g)⇌3Z(g) Kp = 3.18×10−2 at a temperature of 109 ∘C . Calculate the value of Kc. Express your answer numerically.
Question 4 Consider the vector field F(,y)(r,y). (a) Calculate div(F) and curl(F). (b) Is F a gradient vector field? If yes, find f such that F= ▽ (c) Find a low line for F passing through the point r(1) (1,e) 3 4 5 6 8
Question 4 Consider the vector field F(,y)(r,y). (a) Calculate div(F) and curl(F). (b) Is F a gradient vector field? If yes, find f such that F= ▽ (c) Find a low line for F passing...
4. Use Stokes' Theorem to evaluate F dr. F(x,y,z)-(3z,4x, 2y); C is the circle x2 + y2 4 in the xy-plane with a counterclockwise orientation looking down the positive z-axis. az az F dr-JI, (curl F) n ds and VGy, 1) Hint: use ax' dy
3. Consider the vector field F(x, y) + 2y F dr, where C is the circle (r-2)2 +y2 = 1, oriented counterclock (a) Compute wise (Hint: use the FT of line integrals. We could not use it for the circle centered at the origin, but we can use the theorem for this circle. Why?) (b) Let 0 be the angle in polar coordinates for a point (x, y). Check that 0 is a potential function for F
3. Consider the...