Question 3 Let E be the solid enclosed by the ellipsoid x² + + 4 9 1. Calculate the Jacobian using the change of variables x = u, y = 2v, and z = 3w. J(u, v, w) a(x, y, z) (u, v, w) z dV = 2. Rewrite the integral using the change of variables: JIJ [ f(u, v, w) du du dw, where B is the solid enclosed by the unit sphere above E B the uv-plane. Find f(u,v,w). f(u,v,w) = SSS E 3. Evaluate the integral (you may use spherical coordinates to evaluate the integral). = fff f(²₁₂² B z dV = 1 above the xy-plane. (express your answer in terms of u, v, and/or w) v, w) du dv dw= π 5TT

Q: 20. Let S be an oriented surface parametrized by r(u,v). Define the notation do = r, du × r, dv so…

A:

Q: Q4. What is the Laplacian of the following vector function? v = x²x+ xz°ŷ +y° î

A:

Q: Find the Jacobian ∂(x, y) ∂(u, v)for the indicated change of variables. x = 3uv,  y = 2(u − v)

A: on solving this we get the

Q: (5) Use Stokes' Theorem to find the line integral |(* + 2y°)i + (y + z°) +(= + 2r*)k) • dř where C…

A:

Q: Q.14 The integral of the vector A(p, p, z) = "coso ô (standard notation for cylindrical coordinates…

A:

Q: Find the flux of F (x, y, z) = 4.xi+ 4yj+ 2k outward (away from the z-axis) through the surface cut…

A: The field lines of a vector field F through surfaces with unit normal n, the angle from n to F is θ.…

Q: Evaluate . F dr where F(x,y,z) = (cos x, -2x, y + sin(z³)) and C is the boundary of the triangle…

A: This is a problem of surface integral.

Q: 2. The vector function Ř(u, v) = (sin u cos v, sin u sin v, cos u), (u, v) E [0, 7] × [0, 2m] gives…

A: Here, the parametric vector function for the surface is given by R→u,v=sin u cos v, sin u sin v, cos…

Q: Let F= r²i+ (2xy + x) j+zk. Let C be the circle x² + y² = 1 and S the disk a² + y² ≤ 1 within the…

A: (4) Given that , F=x2i+2xy+xj+zk

Q: i. Find b × c in terms of λ . ii. Show that a · (b × c) is independent of λ .  iii. Find the…

A: i)

Q: 10. Use Stokes' Theorem to evaluate ||(V×F)· dS where F= (z,2x, y) and Sis the hemisphere of x' + y?…

A:

Q: pla answer this ASAP,thx Calculate the vector integral(on hte image below) where,F = x 2i + y 2 j ,…

A:

Q: Find the flux of F(x, y, z) = xi+yj - zk outward through the shpere x² + y² + z² = 4. (Hint:…

A:

Q: Let F = –9zi+ (xe*z – 2xe*² ) }+ 12 k. Find f, F •dÃ, and let S be the portion of the plane 2x + 3z…

A: Given, F→=-9zi^+xexz-2xex2j^+12k^ and S be the portion of the plane 2x+3z=6 0≤y≤4

Q: :) Use Stokes' theorem to evaluate RF dr where C F = z?i + yj + xk and C is the triangle with…

A:

Q: Use Stokes' theorem frotF. ds to evaluate F(x, y, z) = (x + tan¯¹yz)i + y²zj + zk, S is the part of…

A:

Q: Use Stokes' Theorem to evaluate ſc F. dr. C is oriented counterclockwise as viewed from above. F(x,…

A:

Q: Evaluate the line integral F.dr. where F(x, y, z) = (5x² + 6y) î — (3x + 2y²)j + 2xz²k, and C is…

A:

Q: 2. Calculate the Jacobian of the coordinate change (x, y, z) to cylindrical coordinates (r, 0, z).

A: To calculate the Jacobian of the coordinate change  to cylindrical coordinates .

Q: The surface S is parameterized by r(u, v) = (u, 2 cos(v), 2 sin(v)), 1 ≤ u ≤ 3,0 ≤ v≤n. 1. Identify…

A: