The necessary and sufficient condition that a velocity potential exists is that curl of V is indentically zero divergence of V is indentically zero gradient ofthe magnitude of V is indentically zero gradient of V is indentically zero

Q: Calculate the work done by a particle moving under the influence of a force F = -(2z + y)ê + x²yŷ +…

A: The general form of a vector-valued function in three dimensions is given by F=F1x^+F2y^+F3z^…

Q: Calculate the flux of F on the boundary of E (oriented outwards)

A: In this question, we have given a closed surface bounded from above by sphere and bounded from below…

Q: A quarter cylinder of radius 2 and height 5 exists as shown in the figure below. Surface integral of…

A: Given that the quarter cylinder is of radius 2 and height 5. This means that the radical distance…

Q: The gradient of a scalar field V is a vector that represents both magnitude and the direction of the…

A: Given : The gradient of a scaler field V is a vector that represents both magnitude and direction…

Q: Find the gradient vector field for the scalar function f(x,y)=sin(2x)cos(6y).   Enter the exact…

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Q: A small surface S is divided into three pieces of area 0.2. Estimate j L F · dS if F is a unit…

A: Lets consider the given surface S  is of the form with area 0.2, where F is a unit vector field…

Q: Show that the outward flux of a constant vector field across any closed surface to which the…

A: We have to Show that the outward flux of a constant vector field across any closed surface to which…

Q: : Find the divergence of the vector field At the point (2,4,1). F(x, y, z) = x² √yi+j+zk

A: We have to find the divergence of the vector field F(x, y, z)=x2y i^ +yxj^+zk^ at the point (2, 4,…

Q: Find the gradient vector field of f(r, y) = In(x + 3y)

A: Solve for the gradient vector

Q: Find the divergence of the vector field F(x, y,z) = x²zi – 2xzj + yzk and the point (2, -1, 3).

A: Divergence of a vector field: The divergence of a vector field F = (P, Q, R) is given by, The given…

Q: A pasticle depicts ts motion in Such a ulay at time ť is P=xîty where, 2= acoskt and y=bsinkt. that…

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Q: Sketch the vector field F(x, y) = xj, the line segment from (1, 4) to (7,4), and the line egment…

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Q: A particle moves along line segments from the origin to the points (1,0,0), (1, 2, 1), (0, 2, 1) and…

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Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y) =…

A: It is required to find the conservative vector field for the potential function given in the…

Q: A vector is said to be conservative when it's: curl equal to zero Laplacian equal to zero divergence…

A: We know that ,  A conservative vector field , is a vector field  F  whose line integral  ∫C F·ds…

Q: ..Find the Gradient Vector Field for: f (x,y, z) = x²ye/z

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Q: Let = (xz, 2 4xyz, rz) be the velocity field of a fluid. Compute the flux of i across the surface…

A: Flux of velocity about a surface is the measure of amount of velocity of fluid hitting the surface.…

Q: The continuity equation for incompressible fluids states that the divergence of the velocity vector…

A: The given statement is The continuity equation for incompressible fluids states that the divergence…

Q: Find the gradient vector field of f бу f(x, у, 2) = X COS Z (бху) (6y бу -sin бх -sin бу i cos k…

A: Given,

Q: 4. Find the flow of the velocity field F = V(xy²z³) along the line segment from (1,1,1) to (2,1,…

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Q: FIND THE DIVERGENCE OF THE VECTOR F GIVEN = sin (8x)i + cos (5y); + z ² k FIELD F(X, Y, 2) BY

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Q: Find the flow of the velocity field F =(x + y)i - (x2 + y2)j along each of the following paths…

A: The velocity field F=(x+y) i -x2+y2j along each of the following paths from 1,0 to -1,0 in the xy…

Q: The directional derivative of the field u(x, y, z) = x² – 3yz in the direction of the - vector (i +…

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Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(4x + y + 62) . F(1, y, 2) =

A: we have to find the gradient vector field of f

Q: The motion of a liquid in a cylindrical container of radius 2 is described by the velocity field…

A: Given,F(x,y,z) = -19y3i + 19x3j + 7k⇒curl F = ijk∂∂x∂∂y∂∂z-19y319x37          = 0i + 0j +…

Q: A scalar field is given by f = x2/3 + y2/3, where x and y are the Cartesian coordinates. The…

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Q: Find the gradient field associated with the function φ(x, y, z) = xyz.

A: To find the gradient field associated with the function  φ(x, y, z) = xyz.

Q: If the components of a vector field F have continuous second partial derivatives and S is the…

A: Given that The surface S is simple and F is twice differentiable To show ∫∫scurlF.ds=0

Q: Find work done by a force field F= in moving object along the line segment from the point (2,1,3) to…

A: Given, F=1y-x,xyz,ex2+y2+z2

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(x + 3y + 4z) . F(2, y, 2) =

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Q: Any charge configuration above an infinite ground plane may be replaced by an arrangement composed…

A: Hey all these terms are being defined

Q: How is the circulation of a vector field on a closed smooth orientedcurve calculated?

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Q: Flow curves in the plane Let F(x, y) = ⟨ƒ(x, y), g(x, y)⟩ be defined on ℝ2.

A: The vector field Fx,y=fx,y,gx,y.

Q: Find the derivative of g in the direction of a = 2i+ curl w at point (0 where น u+2 8 = In- J u =…

A: Given: the function g=lnuu+2, u=z(xy+z2)2, w=cosxyzi+xeyz+1j+xyzk To find: the directional…

Q: A smooth vector field F has div F(5, 2, 5) = 2. Estimate the flux of F out of a small sphere of…

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Q: Determine the gradient of the give scalar field V = e(2*+3y) cos 5z at R(-0.1, 0.2, 0.4).

A:

Q: Let W be the region between the sphere of radius 4 and the cube of side 1, both centered at the…

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Q: Find the gradient vector field of f(x, y) = x° sin(9y). F(x, y) = x°i+ sin(9y)j F(x, y) = 9x°…

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Q: Find the gradient vector field (F(r, y, z)) of f(x, y, z) 5zy ze F(1, Y, 2) =

A: Gradient vector field F→(x,y,z) of a function f(x,y,z) is given by,…

Q: Use the Divergence Theorem to evaluate the flux of F = yi - xj + 2xk across the sphere x ^ 2 + y ^ 2…

A: Given: Fx,y,z=yi-xj+2xk  we want to calculate flux of Fx,y,z=yi-xj+2xk across the sphere x2+y2+z2=25…

Q: Sketch the vector field F(x, y) = xj, the line segment from (1, 9) to (6,9), and the line segment…

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Q: Find the gradient vector field (F(r, y, z)) of f(x, y, z) = x² sin(3yz) F(z, y, 2) = (

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Q: A smooth vector field F has div F(1,4, 4) = 7. Estimate the flux of F out of a small sphere of…

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Q: Let divF(x, y, z) 4x3 + 4y3 + 4z³ , where F represents the flow of a vector field. Estimate the flux…

A: Given that a small sphere of radius 0.1, centered at the point  (1, 1, 1).And                  div F…

Q: Which graph represents the gradient vector field of f(z,9) = Sz + y

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Q: Show that there is no vector field G = (P,Q, R) where P, Q and R have continuous second order…

A: Recall a result, If G= (P,Q,R) is a vector field on IR3 and P, Q and R have continous second order…

Q: (a) Determine the divergence and curl of F. (b) Find all possible potential functions for F R3 on

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Q: Calculate of A= Xzi- 2x*yzj+2yz'K at ( 1,-1,2). the Curl and Divergence

A: Given A = xz3i-2x2yzj+2yz4k Compare it with A = Axi+Ayj+Azk Hence Ax = xz3Ay = -2x2yzAz = 2yz4 Then…

Q: Prove that divergence of vector potential 'A' is always zero.

A: