Calculate the circulation of the flow v=-8xy i+4y j around the triangle with vertices (0, 0), (3, 0), (3, 2).

Q: Find the flux of F(x, y, z) = 4а, 32 + 2?, across the positively oriented surface 2 S given by R(u,…

A:

Q: use Green's Thearum. to fina the counterclockwise Circulation. and outward Flux for the flela F and…

A:

Q: ☛ F(x, y, z) = yi + xj + z´kis a vectorial field a) Calculate the flow through the solid formed by…

A:

Q: Q) S = the part of the paraboloid z = 10-x^2-y^2 above z = 1 F = [x^3, xy, z] Find the flux of F…

A: Step:-1 Given that S is part of paraboloid z=10-x2-y2 above z=1 Since, given that z=10-x2-y2 and…

Q: A uniform magnetic field B has constant strength b teslas in the z-direction [1.e., B = (0, 0, b) 1…

A: Stoke's theorem ∫CF→.dr→=∫∫S curlF→.dS→ Where S is a smooth surface bounded by a smooth closed curve…

Q: Find the flux of the field F(x, y, z) = z2 i + x j - 3 z k outward through the surface cut from the…

A:

Q: Use the divergence theorem to find the outward flux of F across the boundary of the region D. F =…

A:

Q: Let v = (x*z, 2 – 4xyz, xz) be the velocity field of a fluid. Compute the flux of v across the…

A: Given, v→=x2z,2-4xyz,xz225x2+16z2=y-202 Then, calculating div v→ div…

Q: A fluid motion is given by v=(y+z)i+(z+x)j+(x+y)k show that the motion is irrotational and hence…

A:

Q: Find the flux of the vector field F (x, y, z) = (e-y, -y, x sin z) across the positively oriented…

A:

Q: A- Use Greens theorem to evaluate the circulation or flux for the velocity 1 y) = x+y j across or…

A: The flux form of Green’s theorem relates a double integral over region D to the flux across boundary…

Q: Find the flux of F = 2xi + yj+ zk upward through the surface r = u?vi + uv²j+ v³k, (0 < u< 1,0 < v <…

A: We will find out the required value of flux.

Q: The outward flux of F(x, y, z) = xi + yj + zk across any unit cube is.

A: Given flux is F=xi+yj+zk. Compute div F. divF=∂∂xx+∂∂yy+∂∂zz=1+1+1=3

Q: Find the flux of the vector field F = (z, y°, through the helicoid with parameterization r(u, v) =…

A:

Q: 4. The vector field F(x, y) = (x² – y²)i+ 10-Rj-2zrk represents the velocity of a gas flowing in…

A:

Q: (x²z, 2 – 4xyz, xz²) be the velocity field of a fluid. Compute the flux of v across the surface 4x2…

A: Given velocity field.

Q: Let div F = 4(8 – z) and 0 < a, b, c < 11. ) Find the flux of F out of the rectangular solid 0 sa<…

A: PART A According to the divergence theorem, Flux=∫x∫y∫zdiv F dV=∫x=0a∫y=0b∫z=0c4(8-x)…

Q: Use the Divergence Theorem to find the flux of F(x, y, z)=z³ i-x³j+y³ k across the sphere x² + y² +…

A: This is a problem of vector calculus.

Q: Geometrically represent the velocity field of a fluid given by i(x, y) = yi – xj, , note that (xi +…

A:

Q: Compute the flux of F (x, y, z) = (y, z, x) across the surface S : R(u, v) = (u+2v, v – u, u), where…

A:

Q: Find the flux of F through the surface S. F(x, y, z) = xi + yj + xzk S is the cone r(u, v) = ⟨u…

A: flux related problem.....

Q: An Eulerian flow field is described in Cartesian coordinates by V = 4i+xzj+5y3tk. (a) Is it…

A: According to bartlrby guidelines we have to answer only first three subparts please upload the…

Q: 4. Find the flux of F = across the curved sides of the surface S = {(r, y, z)|z = cos y, |y| < T,0 <…

A: we have F→=e-y,2z,xy Surface S=x,y,zz=cosy,y≤π,0≤x≤4 Let z=gx,y=cosy ∂g∂x=0,∂g∂y=-sinyy≤π…

Q: Find the flux of F = 2xi + yj + zk upward through the surface r = u?vi + uv² j + v³k, ( 0 < u< 1,0 <…

A: To find the flux of F→ =2xi^ +yj ^ +zk^upward through the surfacer→ = u2vi^  +uv2j^ +v3k^0≤v≤1 ,  0≤…

Q: Find the flux of the field F(x, y, z) = (x, y, -2z) through the surface z = ! =1– 2² – y² (0 < ¤< 1,…

A:

Q: Use Gauss's Law to find the charge enclosed by the cube with vertices (±4, ±4, ±4) if the electric…

A: Given:

Q: A fluid motion is given by V=(y+z)i+(z+x)j+(x+y)k. Is this motion irrotational? If the velocity…

A:

Q: Let C be a triangle with verticies (0, 0), (2,0), and (0, 3) oriented in the counterclockwise…

A: We have to find the outward flux.

Q: Find the flux of F = z²k upward through the part of the sphere æ² + y² + z² = a² in the first octant…

A: Given vector field is  F=z2k to find the flux through sphere x2+y2+z2=a2 in the first octant.

Q: The a and y components of a fluid moving in two dimensions are given by the following functions:…

A:

Q: Find the outward flux across the domain D xi + yj + zk x2 +y? + z? F(x,v,z) = across the sphere x2…

A:

Q: 2. Calculate the circulation of the flow v=-8xy i+ 4 yj around the triangle with vertices (0, 0),…

A: Given that: v=-8xyi+4y2j The triangle is with the vertices (0,0), (3,0), (3,2) To calculate the…

Q: Compute the net outward flux across the sphere r²+y²+z (r*, y², 2ª).

A:

Q: Use the Divergence Theorem to find the flux of F(x, y, z) = 3x°i+ 3.x²yj+ 33xyk across the surface…

A: According to the given information, it is required to use the divergence theorem to find the flux.

Q: F = 2xi+2yj+3zk and σ is the portion of the cone z=sqrt(x^2+y^2) between the planes z=3 and z=5…

A:

Q: 6. (a) Determine whether the given velocity vector of a fluid is rotational or irrotational. I) F(x,…

A:

Q: Find the outward flux of F = across the boundary of D: the upper region cut from the solid sphere…

A: To find the outward flux use divergence theorem

Q: Use the Divergence Theorem to calculate the outward flux of F = (r3 + y, y + z3, z3 + 23) across S:…

A:

Q: A uniform magnetic field B has constant strength b teslas in the z-direction [i.e., B = (0, 0, b)]…

A: We have to verify that A =12B×r is a vector potential, where r = x,y,0     Given that B = 0,0,b…

Q: Show that polar form of equation of continuity for a two dimensional incompressible fluid is av…

A:

Q: Find the potential function for the flow with velocity field i- (2ry + 2ze2*)j- [2ye 2y + z sin (2)…

A: Given:  u=y-2+2xi-2xy-3+2ze-2yzj-2ye-2yz+zsinz-coszk Let u=∇f…

Q: Use the Divergence Theorem to ding the rate at which a fluid with flux F= through the cube formed…

A: Given vector field is F=x2,y2,z2. Gauss divergence theorem:  Let F→=fi+gj+hk be a vector field with…

Q: (a) Consider the flow i (Vr, Vy) = (y², x). Determine the circulation T = ū • dl about the square…

A:

Q: 8. Use the Divergence Theorem to find the outward flux of F(r, y, 2) = (y² + 2) i+y²j+(e® + z) k…

A:

Q: A hot air balloon is shaped like a truncated sphere like the figure. The gases escape through the…

A: Given a hot air balloon has a truncated spherical shape. The velocity vector field  Vx,y,z=∇×Fx,y,z…

Q: (x*z, 3 – 4xyz, xz") be the velocity field of a fluid. Compute the flux of v across the surface 16x?…

A: Use the Divergence Theorem to calculate the flux. Partial differentiate the flux and substitute the…

Q: Use the Divergence Theorem to evaluate N ds and find the outward flux of F through the surface of…

A: We have to find the flux

Q: Find the flux of the vector field F = (y,- z, z) across the part of the plane z = 4 + 3z + 2y above…

A:

Q: Calculate the flux of F across G. F(x, y, z) = - yi + xj; G is the part of the plane z= 20x - 10y –…

A:

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: