Find the null space of the linear transformation T: R →R TE, y, z) = (x + y, z). defined by

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: Let B be the region in the first quadrant of the ry-plane bounded by the lines x + y = 1, x + y = 2,…

A: Solution:-     using transformation:- u=x+y; v=x-yx=u+v2 and y=u-v2x+y=2→u=2x+y=1→u=1…

Q: Find the divergence of the vector field F(x,y,z) = (ye*,2x + 3y, ycosx).

A: Consider a vector field is defined as, Fx,y,z=f,g,h The divergence of the vector field is defined…

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

A: Given that, g(x)=xy+yz+xz   We have to find, Gradient Field.   As we know,  grad…

Q: Sketch the region onto which the sector r< 2, t/4<0Sa/2 is mapped by the transformation w= = iz?.

A: 5) In this question, we have sketch the region in the w-plane by using the region in the z-plane.

Q: V.(V×F)= 0

A: Now given, ∇.∇×F=0 To prove the above.

Q: Give an example of a rigid motion T in C n, T(0) = 0, which is not a linear transformation.

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Q: Find the Jacobian for the transformation x = 3u + v and y = u+ 3v. NEXT, use this transformation to…

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Q: Find the conservative vector field for the potential function h(x,y,z)=3xyln(x+y) by finding its…

A: Find gradient

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = tan(6r + 2y + 2) . F(2, y, 2) =

A: Find the gradient vector field

Q: a) T: R? → R? first performs a horizontal shear that transforms e, into ez – 2e, and then reflects…

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Q: Exercise 10.2 Evaluate the line integral fc(² – 2y²) ds, where C is the line segment from (0,0) to…

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Q: Find the gradient vector field of f(x, y) ,3

A: Given,          The gradient vector field of fx,y=x6y3.

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: Find the divergence of the vector field v = (x²y² - z3)i + 2xyz j + e*yzk. a) 2xy2 + 2xz + xy e*yz…

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Q: Consider the vector-valued function [f(x, y) xt – xy² [g(x, y) 2 Vx + y³x² [f(x, y) Which of the…

A: Find the Jacobian of the given matrix

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: Find the net outward flux of F = a x r across any smoothclosed surface in ℝ3, where a is a constant…

A: Net outward flux is ∇.F ∇=i∂∂x+j∂∂y+k∂∂za=a1i+a2j+a3kr=xi+yj+zk

Q: Carry out the affine transformation | u = (x + y)/V2 lv= (y - x)/V2 in the improper double integral…

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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: Find the divergence of the vector field F = < yx², xzª, zy³ div F =

A: f = y x2  ∂f/ ∂x = 2xy   g = x z4  ∂g/ ∂y = 0   h = z y3  ∂h/ ∂z = y3

Q: Let f(x, y, z) = 3xyz + y?z. Find the gradient field Vf.

A: Formulae of gradient field is given below

Q: Find the gradient vector field of f(r,y, z) = x*y* +y³z³ + æz°.

A: The gradient vector field of f(x,y,z) is given by ∇f=∂∂xf(x,y,z) i^+∂∂yf(x,y,z) j^+∂∂zf(x,y,z) k^…

Q: Use the transformation u = y - x, v = y, to evaluate the integral on the parallelogram R of vertices…

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Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y) = x²…

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Q: Find the divergence of the vector field F. F(x, y, x) = i+ zj - 8y²zk %3D O2+ 7- 8y O zi+ 7zj - 8y²k…

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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: 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 potential function for the vector field given as 8. F(x, y, z) = 4 O a. $(x, y, z) = x³y³z²…

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Q: Find the gradient vector field (F(r, Y, 2)) of f(x, y, z) = In(4x + 6y + 22) . F(7, y, 2) = (

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Q: Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous…

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Q: Find the gradient vector fields for the following potential functions (a) f(x,y) = x²y – xy?

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Q: The vector field F = (y², 2xy + e³2, 3ye³z + x) is conservative

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Q: Given the joint PDF of X and Y: f(x, y) = e-(*+y), x> 0, y> 0; 0 otherwise. Let U V = X + Y. Using…

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Q: Consider the linear transformation T : S R where T(u, v) = (x,y) is given by u = x+y and %3D v = x-…

A: We have to use the transformation u=x+y and v=x-y to evaluate the double integral ∬R x-yx+y+22dxdy.…

Q: Find the gradient vector field ∇f of f(x, y) = x2 − 4y.

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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: Given F = xyi + 1(x – y) j+ 1(x + y) k, (1) Compute the fundamental cross product F, xF, =

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

Q: Find the gradient field that corresponds to function (x, y, 2) y x +x e-y z

A: The given function ϕx,y,z=y2-x2+xez2-y3z The gradient of a function is given by ∇f=∂f∂x,∂f∂y,∂f∂z

Q: Show that the vector field F(x, Y, z) = (x² + y²)i – 2xyj is not conservative on R³.

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Q: Use the Fourier transform to solve the IBVP U = a² (urz + Uyy), 0 < x < 0, 0 < y < o. u (x, y,0) = f…

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Q: Let F(x, y) = (2x + y³, 2y – x4). Compute the flux oF. F.n ds of F across the circle of radius 5…

A: Given function can be written as, Region bounded by the circle of equation,

Q: the function T:C+C defined by T(2) = z a linear transformation ?What if we consider C as a vector…

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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: Let B be the region in the first quadrant of the xy-plane bounded by the lines r + y = 1, x + y = 2,…

A: Answer is mentioned below

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

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Q: Use the transformation (change of variables) x = 2u +v and y = u+ 2v to set up the integral / (x –…

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Q: Consider the vector field F(x, y, z) = (-2y, –2x, 8z). Show that F is a gradient vector field F = VV…

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Q: Consider the region R in the ry-plane that is described by the intersection of the four lines: 6x –…

A: First of all, use the transformation given.