Let f(x,y, z) = a² + 2y² +z² and g(x,y,z)=z-xy. Find a unit vector u satisfying D.f (1.1,2)= 0 = D, g(1.1.).

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

A: The surface integrals compute the flux of fields across surfaces,                   So,…

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

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Q: c) Show that the vector field W(x, y, z) = (2yz+ xz – x², –2xz – yz + y?, y² – x² + z²) - - is a…

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Q: The given vector function is F(x, y, z) = (2x – y + In(z² + 1))î + (x – 6e²y + 4 cos 4y)ĵ + | 2xz -…

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Q: What is the flux of the vector field F(x, y, z) = [9y, 2xy, 4yz + 234xy] through the unit cube [0,1]…

A: We find the flux of the vector field F(x, y, z) = [9y, 2xy, 4yz + 234xy] through the unit cube [0,1]…

Q: Find llFll and sketch several representative vectors in the vector field.

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Q: Find the domain of the vector-valued function. r(t) = n(t) 0 (-∞, +∞0) O (1, + ∞ ) O (0, +∞ ) O [1,…

A: Find the domain for all three components and then take intersection of those.

Q: Find the work done by vector field F(x, y, z) = xi + yj + (x + z)k on a particle moving along a line…

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Q: Find the curl of the vector field F. F(x, y, z) = (2y - z)i + e²j + xyzk

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Q: Find all points (a, b, c) on the sphere x2 + y2 + z2 = R2 where the vector field F = yz2i + xz2j +…

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Q: given the vector function f(t) = (t, e', –2cos 2t) determine a)df/dt f(t)dt

A: The vector function is given by ft = t2, et, -2cos2πt To evaluate: a dfdtb ∫01ftdt

Q: Find the work that is done by the force F = (x + y)î + xyj – z?k acting on a particle that moves…

A: Convert function into parametric function. Then find work done by doing line integration.

Q: Integrate the vector field h (x, y, =) = xy i +yz j+ xz k over r (1) =ui+ử j+ư k, u E[-1,1]. a) 0 b)…

A: According to the given information, it is required to integrate the vector field over the vector…

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

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Q: Determine the work done by the force F"(x,y,z)=(x²y,xy). to move a particle along the line segment…

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Q: Find the gradient vector field of f(x, y, z) = x³y* + y° z³ + xz°.

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Q: Determine if the vector field F=(yz,xz,xy) is conservative?

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Q: Let o = 2 In(r), where r= V x2 + y2. Express Vo in terms of the unit radial vector e, in R2. tep 1…

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Q: Find the work done by the force field  F(x, y, z) = y + z, x + z, x + y  on a particle that moves…

A: Given that the force field,

Q: Let f(x, y) = 6x + 3y2. Find the vector F = Vf. F(x, y) = Evaluate the line integral of F over r(t)…

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Q: Find the gradient vector field of f(x, y, z) 5x+ 2y+8z

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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: Integrate the vector field h (x, y, =)=ry i+yz j+x k over r (1u) = ui+ j+k, -ןue-L1

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Q: Let, T:R' → R;T(x, y, z) = (x + y+ z,2y + z,2 y+3z) verify that [T][v]s = [T (v)]; for any vector v…

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Q: 8) Given z =f (x.y) =1- (x² +4y2) ,Find : a)The vector function, 10 normal to the surface of f at…

A: This is problem of vector calculus.

Q: Find a non-zero and non-constant vector function F(x, y) = F(x, y)i+ F2(x, y)j, %3D OF OF F such…

A: Given, to find a non-zero and non-constant vector function in the form…

Q: Consider the vector function r(t) = (t, t°, 1) Compute r' (1) = ( T(1) = ( r" (t) = ( r' (1) x r"…

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Q: Find a function f (x, y, z) such that ∇f is the constant vector〈1, 3, 1〉.

A: It is given that, ∇f=<1,3,1>   It is known that a vector ai^+bj^+ck^ can also be represented…

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: Find a non-zero and non-constant vector function F(x, y) = F1(x, y)i+ F2(x, y)j, ƏF OF 02F and dy?…

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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: Find the work done by the force field F(x, y, z) = (y + z, x + z, x + y) on a particle that moves…

A: Given To find the work done by the force field Fx,y,z=(y+z,x+z,x+y)  on a particle that moves along…

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

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

Q: Find the gradient vector field (F(x, y, z)) of f(z, y, z) = e®2z + $y + 2= F(1, y, z) =

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Q: -dt. dF = (dr · V)F+ at

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Q: Find the work done by the force F = xy i + ( y - x)j over the straight line from (1, 1) to (2, 3).

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Q: Evaluate the line integral F*dr, where F(x,y,z) = (3sin(x), -4cos(y), 4xz) and C is given by the…

A: Find your answer below

Q: x – y T x + Y. a) A vector in im(T) that is not 0 is

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Q: 5. If a vector function F depends on both space coordinates (x, y, z) and time t, show that dF = (dr…

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

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Q: Find a unit normal vector to the surface f(x, y, z) = 0 at the point P(4, – 5, – 5) for the function…

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Q: Let f(x, y, z) = x² – y? + 2z. Compute the rate of change of f in the direction of the vector (v2,…

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Q: Find the gradient vector field of f(x, y, z) = e® 5x+2y+8z

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Q: Let, T:R' → R';T(x, y,z) = (x – y,2x + 3y + 2z,x+ y+2z) . verify that [T][v], = [T (v)]; for any…

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Q: Show that on the saddle surface z= xy the two vector fields (Vī+x² ± i+ y°, yvī+x² ±xvI+y°) are…

A: Let f(x,y,z)=xy−zf(x,y,z)=xy−z , (a,b,c)∈the saddle surface, and calculate the total derivative…

Q: Find the work done by the force field F(x, y, z) = (x – y², y – z², z – x2) on a particle that moves…

A: We have to find the work done by the force field F(x,y,z)=<x-y2, y-z2, z-x2> on a particle…

Q: Find the gradient vector field of f(x, y, z) = e9z+6y+7z V

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