30. Let D be the image of R = [1,4] × [1,4] under the map G(u, v) = (u² /v, v²/u). (a) Compute Jac(G). (b) Sketch D. (c) Use the Change of Variables Formula to compute Area(D) and I| f(x, y) dx dy, where f(x, y) = x + y.

Q: Compute Div(F) and Curl(F): F--(4y-1) i+yj+(-3y) k

A:

Q: Let B be the region in the first quadrant of the ry-plane bounded by the lines x + y = 1, x + y = 2,…

A: Solution:-     using transformation:- u=x+y; v=x-yx=u+v2 and y=u-v2x+y=2→u=2x+y=1→u=1…

Q: 4. Calculate the integral (x – 3y) dA where R is the triangular region with vertices (0,0), (2, 1)…

A:

Q: Calculate the line integral f -y²dx + xdy where ŕ is the path y² = 2x − x² such that x>1 and y > 0.

A:

Q: 3. Show that the line integral 4. x + y)dx + dy C is independent of path and evaluate this integral…

A:

Q: 2. Evaluate f (3x + 4y2) dA, where R is the region in the upper half-plane 15 bounded by the circles…

A:

Q: 16. Let B be the region in the first quadrant of the xy-plane bounded by the lines x + y = 1, x + y…

A: Given that the region B is bounded by the lines, x+y=1, x+y=2, x=0 and y=0. The given…

Q: 1. Use the transformation u = %, v = xy to find || x y³ dA over the region R in the first quadrant…

A:

Q: a¹y¹2'dadydz, where V is the (6) Using suitable transformations, evaluate closed region in the first…

A: Given the integral  ∫∫∫Vx4y4z4dxdydz Where V is the closed region in the first octant bounded by the…

Q: Find the Jacobian for the transformation x = 3u + v and y = u+ 3v. NEXT, use this transformation to…

A:

Q: and y = 2u, evaluate the integral xy³dA where c) Using the transformation x =-; 6u region R is…

A:

Q: Set up the line integral ∫c x2z ds, where c is the line segment from (1, 6, -1) to (- 4, 1, 5). (No…

A:

Q: Let M be the region bounded by the parabola y = r² and the line y = x + 2. %3D 6. y = r +2 5. 2,4)…

A: Given that, The shaded region bounded by the parabola y=x2 and the line y=x+2 is M.   And the region…

Q: 1. Use an appropriate transformation of coordinates to evaluate xy dA where the region D is enclosed…

A: Special transformation is used to calculate double integral. Given integral is enclosed with the…

Q: 4. Use the transformation u = r - y and v = 2x + y to evaluate the integral I| (22 – ry - y²) drdy…

A:

Q: The area bounded by the curve y3a and the lines z 4 and a-9 above the z-azis is aquare units O 27v0…

A:

Q: The area bounded by the curve y= 3va and the lines z = 3 and T = 8 above the z- aris is square units…

A:

Q: 6. Let D be the region in the ryz-space defined by the inequalities 1 <z< 2, 0 < ry < 2, and 0 < z <…

A: Given integral, ∫∫D∫x2y+3xyzdxdydz where  1≤x≤2, 0≤xy≤2, 0≤z≤1 and transformation u=x, v=xy, w=3z…

Q: 5. If F=2xi - xj + y*k, evaluate ffLF av over the region bounded by the surfaces x = 0, y = 0, y = 6…

A: Given: F→=2xzi∧-xj∧+y2k∧region bounded by the surface x=0,y=0,y=6 and z=x2 ,z=4

Q: Evaluate the integral ∫c(2x-y)dx+(x+7y)dy along the path C, defined as y=9-x2 from (0,9) to (3,0)

A: Given ∫c2x-ydx+x+7ydy  where y=9-x2  from 0,9 to 3,0 Parametrize C by xt,yt=t,9-t2 and 0≤t≤3

Q: Let F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that…

A: Given F={-z2,2zx,4y-x2}. Curve in the plane (s) = x+y+z=4. Area enclosed is 16.

Q: Consider the region RR in the xyxy-plane that is described by the intersection of the four lines:…

A:

Q: 3) Let F = (y+ev=,8x+ cos(y*)) Use a theorem to compute ſ F •dT where c is the boundary of the…

A: Given that  F→=y+ex,8x+cosy3 To compute ∫cF→⋅dr→ where c is the boundary of the planar region…

Q: (z+ y – 1) cos D

A:

Q: 1. Evaluate the line integral f(z, y) ds where f(x, y) = Vx+ 2y and C is the straight-line segment…

A:

Q: 1)Prove that the curves o, = {|z|=2} y 0, = {|z- i|=4} are homotypic. 2)calculate O z*dz and o z*dz…

A:

Q: Evaluate O F.dT where +(3y %3D y + 1 and C is the triangle with vertices: (0,0), (6,0), (3, 5).

A:

Q: Evaluate ∫∫∫E √x2 + z2dV, where E is the region bounded by the paraboloid y = x2 + z2 and the plane…

A: Given, y=x2+z2 ⇒z2=y-x2z=±y-x2 Therefore the limit of z is -y-x2≤z≤y-x2 On drawing the graph in the…

Q: 8. Use Green's Theorem to calculate f, x?y³dx + (xy - y?)dy where C is the boundary of the region…

A: The given integral is ∫Cx2y3dx+xy-y2dy, where C is the boundary of the region lying between the…

Q: Use the transformation u = - -6x +y, to evaluate the integral [ ( (5x • y) dx dy, where R is the…

A: use the transform u=-6x+y, v=5x+ywe evaluate ∫∫R 5x+y dx dywhere R is the parallelogram bounded by…

Q: Evaluate f(x,y,z) dS 2. 3DX²+4y+z on the surface S {(x,y,z):05xS2, y= 3, 1Sz5 4)

A:

Q: 3. Evaluate the line integral y² dx + x²y dy where C' is the rectangle with vertices (0,0), (5,0),…

A: Solution is below

Q: Use Greens Theorem to evaluate: §. (2x + ev´)dy - (4y2 + e*")dx where C is the boundary of the…

A:

Q: Evaluate .n dS and .ndS if F = (x + 2y)i– 3zj +xk, , and S is the surface of 2x + y + 2z = 6 bounded…

A:

Q: Let F(x, y, z) = (x-1z, y¬lz, In(xy). (a) Verify that F = V ƒ, where ƒ(x, y, z) = z In(xy). (b)…

A: Given Fx, y, z=x-1z, y-1z, lnxy (a) Verify for F=∇f where fx, y, z=z lnxy The gradient of a function…

Q: |Evaluate the line integral: f.(r² + y) dx + (4x – cos(y)) dy where C is the boundary of the square…

A: According to Greens theorem, the line integral ∮CPdx+Qdy=∬DQx-PydA. BY the extended version of…

Q: 20. Let D be the region in xyz-space defined by the inequalities 1< x< 2, 0 < xy < 2, 0 << 1.…

A:

Q: Let f(z) = 5z + 14, and let C1 be the line segment from z = 0 to z = 2, let C2 be the line segment…

A: In this question, we calculate the complex integral along the given contour joining vertices of the…

Q: - Let f(z) = cosh z. (1) Find the image of S := {z = x+iye C: x < 0,0 < y < } under f.

A: Follow the steps.

Q: can u help me with pullbacks and integration plz

A: Given that the integral,

Q: Let D be the region bounded by the parabola y = x2 and the curve y = sin x, and let P represent a…

A: If the input of the function is more than one, such functions are called multivariable functions.…

Q: Let f(x) = 2x - 1 and g(x) = x - 5 on [ 5 , 9 ]. Find the center of gravity of the region between…

A: f(x)=2x-1 and g(x)=x-5 on [5,9]

Q: 2. Suppose that the edge lengths r, y, and z of a closed rectangular box are changing at the…

A:

Q: Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following…

A: To determine the rate at which the box’s (a) Volume, (b) Surface area and (c) diagonal length are…

Q: The curl of f(x,y,z)=xyzj is None of these. хуk-yzi О ху-yzk ООО О

A:

Q: Use the transformation (change of variables) x = 2u +v and y = u+ 2v to set up the integral / (x –…

A:

Q: 16. Let B be the region in the first quadrant of the xy-plane bounded by the lines x + y = 1, x +y =…

A: I have Provided this answer in step-2.

Q: Use the transformations u = xy and v=xy* to evaluate ff xyd4 where R is the √√3 1 X region bounded…

A:

Q: Let F = (r-y, z+y+3, z2) and S y-direction. Compute {(r, y, z) : y =1-r²-z², y20} oriented in the…

A: Here orientation is positive y direction, surface is open.  During the calculation we ignore many…