12. Evaluate the surface integral JJ, F ·dS for the vector field F(x,y,z)=(-x,-y,z'), where S is the part of the cone z=x + between the planes z=1 and z=3 with downward orientation.

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: 1. Determine the velocity field v so that the surface F(r, y, t) (y+t)+ (r +t)2 - a 0 to be a…

A: Solution:

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: 4. Find the positively oriented flux over the surface created by the paraboloid z = 16 – x² – y2…

A:

Q: Find the work done by the force field F(x,y) = 4yi + 2xj in moving a particle along a circle x2 + y2…

A: We follow

Q: 2. Find the curl of the vector field. F = z°e* sin(2 y)i – e' cos yj+ xyzk to

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Q: Compute the work done by the force field F =< x² + 2y, 4x + y° where C is the boundary of the region…

A:

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: Find the flux of the field F = y2i + xzj - k outward (normal away from the z-axis) through the cone…

A: Definition:

Q: 4. Compute the outward flux of the vector field F(r) = (r, y', =') through the unit sphere r² + y² +…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find the net flux of the vector field F(x, y, z) = (2x, - 2y, z) over the closed surface enclosed by…

A:

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

A: We will use of gauss divergence theorem

Q: 2.) Calculate the curve integral over the vector field V(7, y) = ( -x*y xy for the positive oriented…

A: Let x=2cos t y=2 sin t dx=-2sint dt dy=2 cost dt

Q: Evaluate the line integral F. dr, with vector field F(x, y, z) = sin xi + cos yj + xz k, and path C…

A:

Q: 3. Consider the vector field F = 2xy i + (x² + 2y) j. Verify the Green's Theorem in the closed…

A: Definition of green's theorem :

Q: 10. [ Find the flux of the vector field F(x, y, z)= 3 over the surface S, where S is the surface of…

A: The general form of a vector-valued function in three dimensions is given by F→=Fx, Fy, Fz, where…

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

A:

Q: 1. Determine the velocity field v so that the surface F(r, y.t) = (y+t)+ (z+t)? - a² = 0 to be a…

A: The given surface is - F(x,y,t) = (y+t)2 + (x+t)2-a2.......(1) The velocity field (V→) components…

Q: Consider the following vector fields: F = (3x²e²ª, 2x³e²y + 2 cos y sin y) and Ġ = (y+z, x+y, z —…

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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: 3. Consider the vector field F(x, y) = (2xy, x² – sin y). Evaluate %3D F. ds where C is the curve…

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Q: Compute the integral of the vector field over r(t) = (cos(t), sin(t), t) for 0sts n. F(x, y, z) =…

A:

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

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Q: Find all potential functions of the conservative vector field F (x, y, z) = (2xz + y², 2xy,x² + %3D…

A:

Q: 2.) Calculate the curve integral over the vector field V(7, y) = ( -x*y xy² for the positive…

A: Green's theorem

Q: Consider the following vector fields: F = (3x² e²y, 2x³e²y + 2 cos y sin y) and G= (y+z, x+y, z —…

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Q: Compute the line integral of the vector field F = (4y, –4r) over the circle z2 + y? = 16 oriented…

A:

Q: 12. Evaluate the surface integral F-dS for the vector field F(x, y,z) =(-x,-y.z), where S is the…

A: The given problem is to evaluate the given surface integral in the part of the cone  between the…

Q: - L ---- Let F be the vector field (r², 2r, z²). Let C be the boundary of the surface S given by 4x²…

A: Given- Let F→ be the vector field x2, 2x, z2. Let C be the boundary of the surface S given by 4x2 +…

Q: 15. Verify Stokes' Theorem for the vector field F = 2xyi + xj + (y + z)k and the surface z = 4 – x²…

A: please comment if you need any clarification. If you find my answer useful put thumbs up. Thank you.

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: If F = xy i+ (x² + 4yz)j+ 2y²z k. Evaluate the line integral between A (0, 2, 0) and B (3, 6, 1).…

A: Given, F= xy i +(x²+4yz)j +2y²zk r0= (0,2,0) r1= (3,6,1)

Q: 2. Find the surface integral SS. ř · dS for the vector field F = xyî + yzĵ + zxk where S is part of…

A: To find Surface integral ∬SF→·dS→: The Surface S is of the form z=g(x,y) and is oriented upwards.…

Q: Suppose I wish to fine the flux of the curl of the vector field F(x,y,z) = vector brackets (-y(x +…

A: Given that the vector field F(x,y,z) with the surface parameterized by  r(t,s)=s cos(t),s…

Q: 11. Compute the surface integral (flux) of the vector field F over parametrized surface S SSFdS…

A:

Q: Confirm that the force field F = xy'i+ x²yj is conservative in some open connected region containing…

A:

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

A:

Q: 6) Given the vector field F = (-x,-y,-z) '(x²+y2+z²)3/2! write the integral to calculate the outward…

A: First we parameterize field F then we calculate the Flux across the sphere using following formula:

Q: 3. Evaluate JF dS over the given surface z = xey caused by the vector field F (xy, 4x, yz) over the…

A: The projection of surface in the xy-plane is D={(x,y):0≤x≤1 and 0≤y≤1}. The equation of the surface…

Q: For the force field F = (2xe¬Y, (2y – x²e¬y)) in some open connected region containing P(1, 0) and…

A: Given that, F=2xe-y, 2y-x2e-y If F is conservative then ∂2F∂x∂y-∂2F∂y∂x=0

Q: 21. Compute the flow of the vector field F = r through the surface of the ellipsoid y² + 2 = 1 in…

A:

Q: Evaluate the surface integral JJS ² F. ds for the given vector field F and the oriented surface S.…

A: Consider the vector function F = x i+y j+8 k Here surface S is the boundary of the region enclosed…

Q: 4. A) Find the work done by the force F (x, y) = yx²i – y²j over the curve r (t) = t’i+t*j, 0<t<1.…

A:

Q: 4- Evaluate the surface integral of the vector field F = [ax, by, cz] over the sphere r² + y² + 2² =…

A:

Q: Compute the flux of the vector field F = zi + 8xj through the parameterized surface S oriented…

A:

Q: Find the work done by the force field F(x, y) = xyi+ x²j on a ,2 particle that moves along the curve…

A: Consider the given force field Fx,y=xyi+x2j Then, F⇀x,y=xy,y Write in parametric form y=t  then…

Q: Find the work done by the force field F(x, y) = (4 e* – y³ î + (2 cos(y) + x³ )j on a particle that…

A:

Q: 03 Verify Stokes theorem for the vector field F= 2(-+x)i-zyi-zy'k over the upper hlf surface of…

A: To find- Verify Stokes theorem for the vector field F→ = 2-y2 + xi→ + z2yj→ - zy2k→ over the upper…