39. Let S be the portion of the cylinder y = e* in the first octant that projects parallel to the x-axis onto the rectangle Ry: 1

Q: 5) Consider the conservative force field F(x, y) = (6x²y? + cos(x), 4x y). a) Give a general…

A: For a vector field F=P,Q that is open and simply connected in a region with continuous first order…

Q: Consider the solid E determined by the inequalities x² + y² + z≤ 4, and let S denote the surface…

A: Here we will use Stokes theorem  to calculate  flux of curl (F) on surface S bounded by curve C…

Q: Evaluate the surface integral ∬SF⋅dS for the given vector field F and the oriented surface S. In…

A: Compare the given equation with general equation and find the value of P, Q and R.

Q: The following figure shows a half-cylindrical closed surface, defined by the cylinder x² + y² = 4…

A: As per the question we have to find the total flux of the vector field V through the flat top…

Q: Compute the flux of the vector field F(x, y, z) = (2y, 3, –2xz) through the surface S, which is the…

A: Given vector field is, Fx,y,z=2y,3,-2xz. The given surface is, y=x2+z2  and  x2+z2≤1. Now for the…

Q: Calculate the circulation of the field F = (2xy,y²) in a counterclockwise direction and the outward…

A: Consider the given field. F=2xy,y2 And, the given curves are defined as, y=x and y=x3 The green…

Q: 9. Compute the flux of the vector field F(x, y) = (-x sin(1+xy) – 4 cos(TY), y sin(1+ xy) + 6x)…

A:

Q: Find the outward flux of the field F = (x + y)î + yĵ + zk across the closed surface: S: z = 16x² -…

A:

Q: Compute the flux SS F.nds through the piece of the cylinder of radius 2, centered on the z-axis,…

A: The given surface is the cylinder of radius 2 centered on z-axis, where 0≤x, 0≤y and 0≤z≤4. The…

Q: From the given vector field F⃗ (x, y, z) = , compute the curl of F⃗ and find the values of a and b…

A:

Q: 29. Let K be the part of the cone z = Vx2 + y? between z = 1 and z 3, oriented upward. Let F(x, y,…

A: Flux through the fluid

Q: 3y 1. Given the vector field F(x, y, z) = · î + Vxz·ĵ+x cos y k, find curlF and div F. %3D 2. Use…

A:

Q: 1. Consider the vector field F(x, y, z) = (2e2" cos(3y), –3e²" sin(3y) + Y a. Find divF(x, y, z) and…

A:

Q: I. Use the Divergence Theorem: (² F. ds = Ifv. Fav SSS E = to calculate the flux of the field F…

A: Divergence theorem uses  Double integration converts into triple integration.

Q: Find the outward flux of the field F = 2xyi + 8yzj + 6xzk across the surface of the cube cut from…

A:

Q: 3. Consider the vector field F = (y, -x, x² + y?). Find the flux of F through S if: 9– x2 – y? above…

A:

Q: Let S be a simple closed surface in R' that encloses a solid region with volume equal to 5. Find the…

A:

Q: Find the circulation and flux for the field F(x, y) = (y – 6x² , x+y²) around and across the…

A: Given data field Fx,y=y-6x2,x+y2 around and across the triangle given by y=0, y=x and x=1.

Q: 2) Given the vector field F = (- Vx²+y² 2z2) write the integral to calculate the outward x²+y² flux…

A: We have to find the outward flux across the cylinder x2+y2=100.

Q: | 23. Find the outward flux of the field F = ( 3xy i + (e* + tan¯ly)j х 1 + y?, across the cardioid…

A:

Q: Find the flux of the field F = x2 j - xz k outward (normal away from the yz-plane) through the…

A:

Q: 23. Find the outward flux of the field i + (e + tan'y)j 1 + y², F 3xy %3D across the cardioid r =…

A: Given: The following formula is used to find the outward flux, Comparing with the general form,…

Q: b) Find the flux of the vector field F(x,y,z) = 3xi – yj + 2zk; across o where o is the portion of…

A: The given question belonging to the vector calculus  in which we have to find the value flux of the…

Q: Let S be surface of the closed region E bounded by the parabolic cylinder æ = y² and the planes z =…

A:

Q: Use the Divergence Theorem to compute the net outward flux of the field F=<x,2y,-z> across the…

A: From the Divergence theorem, for a positively oriented boundary surface (outward Normal) of region E…

Q: Confirm that the force field F = xy²i + x²yj is conservative in some open connected region…

A:

Q: Let i = (a° cos z, 5 – 3a'y cos z – 3yz? sin æ, 2° sin æ) be the velocity field of a fluid. Compute…

A:

Q: Зу 1. Given the vector field F(x, y, z) = î+ Vrz j+x cos y - k, find curl F and div F.

A:

Q: Find the flux of the field F = 4x i + 4y j + 2 k outward (normal away from the z-axis) through the…

A:

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, y, - 2z) across the…

A:

Q: Let E be the solid region bounded by the upper half-sphere x2 + y² + z² = 4 and the plane z = 0. Use…

A:

Q: In(t) i + t312j + t cos(at)k, F. dr of vector field F(x, y, z) = 3x²zi + zj + (x³ + 2yz)k along…

A:

Q: 12. Use a parameterization to find the flux across the surface cut from the cylinder x? + y? = 1 by…

A: Image is attached with detailed explanation.

Q: Find the outward flux of the field F = 6xyi + 4yzj + 2xzk across the surface of the cube cut from…

A:

Q: 4. (a) Show that the outward flux of the postion vector field F(x,y,z) = (x)7+ (y)] + (z) through a…

A:

Q: Find the counterclockwise circulation and the outward flux of the field F = (-sin y)i + (x cos y)j…

A:

Q: 2.4) Use the divergence theorem to find the outward flux of the vector field F(x, y, x) = x³i+ y³j+…

A: The divergence of the given vector field is  ∇·F=∂∂xx3+∂∂yy3+∂∂zz3=3x2+3y2+3z2 Let, S be the surface…

Q: Find the total flux of the field x+ /1+ y² + z² 2y + V1+x²+ z² \3z + /1+x² + y², F(x,y, z) = out of…

A:

Q: Use the Divergence Theorem tO compute the flux oT the field 1 25x cos z, 6xy ) out of the region E…

A: Given,       1z2+y2arc tanxz2+y2, 25xcosz, 6xy out of the region E that lies between the spheres…

Q: Find the outward flux of the field F = 3xz2 i + y j - z3 k across the surface of the solid in the…

A:

Q: Let S be the surface in R' that is the part of the ellipsoid 2x2 + y² + z? = 2 satisfying y > 1.…

A: In this question, we use the div.  theorem to evaluate the surface integral. given that…

Q: Consider the surface integral of the vector field F= [xy², 3x², x²z]over a surface S, where S…

A: From Gauss divergence theorem, we have ∬SF→⋅dS=∭VdivF→dV   The vector field is F→=xy2,3x2,x2z.…

Q: 2) Given that the total flux across the surface, S of the tetrahedron bounded by the planes x = 0, y…

A:

Q: Consider the conservative vector field 4 8 F(x, y, z) = – 3y²cos (3nz),37y sin (3tz) – x2). 2xz, %3D…

A:

Q: 10. Use the Divergence Theorem and the vector field F = to compute the outward flux from the region…

A: Here we will evaluate the outward flux.

Q: Suppose that S is the boundary surface of the solid region B that is bounded by the paraboloid z =…

A: Given vector field F→x,y,z=2zey+xy2i→+xz2-2zexj→+x2z+2xe-yk→. To find the flux of the vector field…

Q: Find the flux of the field F = z k across the portion of the sphere x2 + y2 + z2 = a2 in the first…

A:

Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

A: Flux of the given field is given by

Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

A:

Q: Consider the one-form w = x' cos(y) dx + xe13% dy+4z² dz on R3. The associated vector field is: F(x,…

A: For the solution follow the next step.