Let F = (2z, - x, 3y): Find the flux of F, that is, - dS, where S is the graph of the function z = 3x2 + 5y², 0

Q: Find the gradient field F = Vo for the potential function o below. P(x,y,z) = In (x2 + 2y2 + z²)…

A: We have to find gradient of given function.

Q: Find the flux of F¯ = (x − y)i + xj across the circle x 2 + y 2 = 4 in the xy-plane

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Q: Find the gradient fields of the functions g(x, y, z) = xy + yz + xz

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Q: Use the local linear approximation of f(x, y, z) = In(xz + y) at (2, -1, 1) to approximate f(1.98,…

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Q: 6. In each case, compute the flux (x°, y°, z³). W- (a) F(x, y, z) = and a? + y? + 2² = 9 anc…

A: Use divergence theorem

Q: (a) Find the gradient and Laplacian of the scalar function T(r,0,¢) =r(cos0+sin0cosø)

A: As two questions are given I will solve only the first: We are given: T(r,θ,ϕ)=r(cosθ +sinθcosϕ)…

Q: Find the flux of F = yi + 2zxj – z²k through the parabolic cylinder y = x2,0 < x < 1,0 < z< 4, in…

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Q: Let flxiy) = Rx Jy 27 %3D a) Finid the gradrent of f b) Find the directional derivative of f at the…

A: Consider the function as  fx,y=2xy

Q: Find the gradient field of the function, g(x,y,z) = e 3z - In (x +2y).

A: Given function  gx,y,z=e3z-lnx2+2y2                     (1) The gradient field is  ∇g=∂g∂x,∂g∂y,∂g∂z…

Q: Use the divergence theorem to find the flux of F = [x²yz, -xy² z, 6 – 5z2| on the hemisphere S =…

A: Consider the given function. The divergence theorem is defined as. Now, find the value of div F.

Q: Use the divergence theorem to calculate the flux of F = (x - 8y)i + (y – 4z)j + (z - 8x)k out of the…

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Q: he upward flux of

A: Introduction: Let F be a function and let the region is S, then the flux of the function is given…

Q: A table of values of a function f with continuous gradient is given. Find Vf dr, where C has the…

A: at t = 0, x=02+1 =1 , y=03+0=0  ⇒ Point is (1,0) at t = 1, x=12+1=2,  y=13+1=2 ⇒Point is (2,2)

Q: Find the flux of the vector field F(x, y, z) = (yz, y°, xz) %3D across the surface S, where S is the…

A: Given

Q: Find the circulation of F = 2xi + 2zj + 2yk,r(t) = ti + (1-t)j,t %3D %3D E [0,1]

A: Here we have to find the circulation of      F=2xi+2zj+2yk, r(t)=ti+(1-t)j , t∈[0,1] .

Q: Use the Divergence Theorem to find the flux of F = (xy², x²y, y) outward through

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Q: Calculate the gradient, the divergence and the curl for the following fields, when possible: ₁ = e/y…

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Q: 1. Let F(x, y) = (y, 2x2). Use the circulation form of Green's theorem to calculate the circulation…

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Q: Find the circulation of F over the counterclockwise closed curve C: F= tan-1yi+ + 2x + y+z)j +yk…

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Q: Find the gradient fields of the functions ƒ(x, y, z) = ln√(x2 + y2 + z2)

A: The given function is fx,y,z=lnx2+y2+z2   The gradient field of a function is represented by,…

Q: Compute the flux ∮∂D F • n ds of F(x, y) = {x3, yx2) across the unit square D using the Flux Form of…

A: Given the equation F(x, y) = (x3, yx2)  The closed surface is the unit square the limits of x and  y…

Q: Find the outward flux of F = (4x +25y*,0,) 10z through the surface 25 4 9

A: Solution:-

Q: For the vector Ā =e - yzē, + xzē, and scalar function f(x,y, z) = x² sin(y) + y a) Obtain the…

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Q: Calculate the divergence of F(x, y, z) = xye² i +yze* k

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Q: Let L(r, y) = 2y – x + 3 be the local linear approximation to f(r,y) at (1,1) . Then f-(1,1) = %3D O…

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Q: Q. A-Find the gradient (Vf) for the following level surface at the given point: f(x, y, = x2 + y² -…

A: Given: f(x,y,z)= x2+y2-2z2+zlnx  Gradient,∇f=i∂f∂x+j∂f∂y+k∂f∂z

Q: Letf(x,y,z) = x^2+y^2+z^2.  Calculate the gradient of f.  Calculate ∫_C(Fdr)  where F(x,y,z)=(x,y,z)…

A: Given  f(x,y,z) =x2+y2+z2

Q: Use the divergence theorem to calculate the flux of F = (x - 9y)i + (y – 5z)j + (z – 3x)k out of the…

A: The divergence theorem mathematically states that:

Q: Let E = {(x,y, z) | x² + y² < 1 and xy < z < 2} and let S be the boundary of E, oriented outward.…

A: Using Gauss divergence law:  ∯SF.dS=∫∫∫V∇·F.dV                   =∫∫∫V2+3-1.dxdydz…

Q: 5. Find the flux of F = 2xi + (x- y)j across the circle r(t) = acos[t] i + asin[t]j,t e [0,2n]

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Q: For each of the following vector fields F , decide whether it is conservative or not by computing…

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Q: (a) For a scalar function f(r), show that the curl of the gradient is zero. That is, V (

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Q: For each of the following vector fields F, decide whether it is conservative or not by computing the…

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Q: If f(x,y)=x2+y², then the gradient field at (1,-1) is given by: O. f(x,y)= -2i + 2j O II. f(x,y)=…

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Q: Find the circulation of F over the counterclockwise closed curve C: F= tan y i+(, +x +z)j+yk 1 + y²…

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Q: Let F= (x² + y³)i + (2xy)j•Calculate flux F orientated counterclockwise across the curve C:x2 + y? =…

A: Given  F=(x2+y3)i+(2xy)j To find the flux across C which is  C:x2+y2=9

Q: Let - (xz, 5 - 4xyz, xz) be the velocity field of a fluid. Compute the flux of i across the surface…

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Q: Use the divergence theorem to find the flux of F = [x²yz, –xy² z, 3 + z²] on the hemisphere S = {(x,…

A: Find the flux of the vector field F→=x2yz, -xy2z, 3+z2 on the hemisphere S=x, y, z : x2+y2+z2=4, z≥0…

Q: 4. Calculate the flux of F = over the square shown below, oriented counterclockwise. SS (Px +Q, R…

A: We have to check given work on this problem write or wrong   But this work is wrong Because flux or…

Q: A building bounded by the equations-= i-2²-y". z=0 and x=0 is designed to be half a spherical cap.…

A: Given vx,y,z=xz2,x2y−z3,y2z−110. And a building bounded by the equation  z=4−x2−y2, z=0 and x=0 is…

Q: Use the Divergence Theorem to find the flux of F = xy2i + x2yj + yk outward through the surface of…

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Q: Let E = {(x, y, z) | æ² + y² < 1 and cy < z < 2} and let S'be the boundary of E, oriented outward.…

A: Given: x2+y2≤1  and xy≤z≤2Thus, In cylindrical coordinatex=rcosθ,y=rsinθ, z=zxy=r2cosθsinθ0≤r≤1;…

Q: Consider the following function. f(x, y) = 8y sin(xy) Find the partial derivatives. fy(x, y) = ,(x,…

A: From the given function: fx,y=8ysinxy As we know that; The chain rule: fgx'=f'gx·g'x The product…

Q: Given that u = x? - y? and v = - 2xy, determine the stream function and potential function for the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let p(x,y) = e=*y+12g° and let F be the gradient of y. Find the circulation of F around the circle…

A: Given that:           φx,y=e3x2y+12y4 Let F be the gradient of φ and C be the circle of radius 3 and…

Q: Find the gradient field of the function g(x,y,z) = 8xy + 2yz + xz. Vg= (Type your answer in terms of…

A: We have to find the gradient field.

Q: Find the gradient field of the function g(x,y,z) = 2xy + 5yz + 7xz. Vg= (Type your answer in terms…

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Q: Justify the following statement: The flux of F = (x3, y3, z3) through every closed surface is…

A: The flux through a closed surface is given by Gauss's divergence theorem, Flux, ∫∫∫∇→·F→dV=∯F→·n^ds…

Q: Let if (x, y, z) # (0,0, 0) if (x, y, z) = (0, 0, 0) xyz x²+y²+z² > f(x, y, z) = { 10, (a) Use the…

A: fx, y , z = xyzx2 + y2 + z2, if (x, y , z) ≠(0,0,0)                             0,…