Find the directional derivative of the field f(x, y, z) = -4x³ + 3 xy² + 5 y³ at P = (-1,–3,5) in the direction a = (1,3, –4). Give an exact answer. D(1,3,-4)/(-1,-3,5)

Q: Find the divergence of the field. F = (2x+7y-8z)i + (5x +3y+6z)j + (-6x-y+2z)H div F =

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Q: Find the curl of the vector field F (5x sin(y), 2y cos(æ)) curl F =

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Q: Find the divergence of the vector field.

A: To Find the divergence of the vector field.

Q: Find the divergence of the field. F=(-x+8y - 3z)i+ (8x- 4y + 3z)j+ (- 4x- y- 3z)k

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Q: Which of the following functions is a potential function for the conservative vector field F(x, y,…

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Q: Find the directional derivative of the field f(x, y, z) = In(4 x) at P = (1, –2, –5) in the…

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Q: 10. For a vector field F = <2xy + y³, x² + 3xy²+ 2y7 find a function f such that F = Vr.

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Q: For which value of a the field i = (y –x)i +(2x – y)j + azk is solenoidal. Find the gradient of the…

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Q: Q4 A) Calculate the divergence of the vector field F=y³i + xyj B) Find the Curl of F(xy.z)=3x²i +…

A: To find out the divergence of vector field.

Q: Find a potential function for the vector field F(x, y) = (14xy + 7y- 8, 7z + 7z). f(x, y) =

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Q: 5. Let F be the gradient vector field of the function f (x, y) = 4xy + 3y". Evaluate JaF. dr, where…

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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 of f(x, y) ,3

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

Q: Choose a correct potential function for the conservative vector fieldF = (3x-y+1, –x-4y-2).

A:

Q: Find a potential function for the vector field F by inspection. F = (4x, 4y) P(x, y) =

A: Given:

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: of f. Sketch the gradient field for f (x, y) = /x² + y? along with several level curves

A: Here we have,

Q: Verify if the vector field F = (x° cos(4y), -x sin(4y)) is conservative and evaluate the line…

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Q: Evaluate the line integral of the vector field (3x2 + 3y + 2xy, 3y2 + 3x + x2) along the curve y = x…

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Q: Find the curl of the vector field F curl F = (2y cos(x), 5x sin(y))

A: F = (2ycosx , 5xsiny) find curl F =?   curl F  = dF2/dx - dF1/dy here  F1 = 2ycosx  F2 = 5xsiny now…

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

A: Formulae of gradient field is given below

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: Q3) Determine if the vector field F(x, y, z) = (yz,xz + 2y, xy + 1) is conservative. If it is, find…

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Q: Q1. What is vector field and potential function? Sketch the gradient field of p(x, y) = x² + y² –…

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

Q: Q5) If the vector field T = (axy + Bz³)a, + (3x² – yz)a, + (3xz² – y)a, is irrotational, determine…

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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: Find the gradient field F = Vq for the potential function o = 3x y-y²x. %3D F = O0

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Q: Find the curl of the vector field F = (3y cos(x), 5x sin(y))

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Q: Find the curl of the vector field F = (3x sin(y), 7y cos(x)).

A: The solution is:

Q: Find the line integral of the vector field F= over the curve C which is the quarter of the unit…

A: We have to find line integral from point (0,1) to (1,0)

Q: If F = Pi+ Qj+Rk is a vector field on R and P, Q. and R have continuous second-order partial…

A: Given:F=Pi+Qj+Rk is a vector field  in ℝ3To show that:∇.curl F=0

Q: Use the method of isoclines to sketch the slope field for y =x +y-1, then draw the solutions…

A: In this question, we sketch the direction field for the differential equation y'=y+x2-1 To draw the…

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

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Q: Find the conservative vector field for the potential function h(x,y,z) = 16xy ln(x+y) by finding its…

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Q: Q1. What is vector field and potential function? Sketch the gradient field of p(x, y) = x² + y² -k².

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

Q: Consider the vector field F(x, y) = (x², −y). If (x, y) is in the first quadrant then the vector…

A: Part 1. (x, y) is in the first quadrant.  =&gt; x &gt; 0 and y &gt; 0 =&gt; x2 &gt; 0 and - y &lt; 0…

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: (a) F = (10xz + y) i + 2xy} + 5x² k: curl F = F is a gradient field (b) G = (5xy + yz) i + (5x² + z²…

A: F→=10xz+y2i→+2xyj→+5x2k→ curl F→=i→j→k→∂∂x∂∂y∂∂z10xz+y22xy5x2 therefore F is a gradient Field

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: Find the gradient field F = Vo for the potential function = 2x³y-4y³x.

A: We have to find out gradient

Q: The divergence of a vector field F = 3x² yz? î + 2yz ĵ – xz² k at he point (1,0, 1) is -2 O 2 O 1

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

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Q: Which of the following functions is a potential function for the conservative vector field F (x, y,…

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Q: Evaluate F•Tds for the vector field F = x²i- yj along the curve x= y? from (1,1) to (4, – 2).

A: Answer = 19.5

Q: Consider the conservative vector field given by: F (x, y) = (x - ycos (x), y - sin (x)) A potential…

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Q: If f(x,y) = x^3+y^3, sketch the gradient vector field: \grad{ f(x,y) }

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Q: Q3 Show that the vector field F= 2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

A: Solution

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

A: We have to find the conservative vector field.

Q: Find the conservative vector field for the given potential function. f(x, y) = 9x2 + 2xy + 10y2 F(x,…

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Q: 25. Evaluate |F•T ds for the vector field F = x²i – yj along the curve x = y? from (4, 2) to (1,–1).

A: