Show whether or not F is a conservative field. If it is, find a potential function for it and use it to evaluate the line integral F dr along any path from (0,0) 6. to (1,1). F(x, y) = (2.xy + y)ĩ +(x² + x)}

Q: Show that work performed by a constant force field F over any path C ~ from P to Q is equal to F •…

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Q: 4. If div F >0 in a region enclosed by a small sphere, the outward flux of the field across the…

A: Given: Divergence F>0 is enclosed by a small sphere in a closed surface . The objective is to…

Q: 4. Is this vector field F(x,y)=(ycos x)ĩ +(sin x+ y)j being the path-independent field? Please show.

A: To check if the given vector field is path-independent field or not. Given:

Q: Show that work performed by a constant force field F over any path C from P to Q is equal to F - PÓ.

A: A vector has 2 components, magnitude and direction which is represented by cos and sin angles. A…

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: Compute the flux SS F.nds through the piece of the cylinder of radius 2, centered on the z-axis,…

A: The given surface is the cylinder of radius 2 centered on z-axis, where 0≤x, 0≤y and 0≤z≤4. The…

Q: Prove that F(x, y, z) = (y² cos x + 2*)i+ (2y sin x – 4)j+ (3xz2 + 2)k - is a conservative vector…

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Q: Find the work done between point 1 and point 2 for the conservative field F. F= (y +z)i + xj + xk;…

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Q: 13 Is this vector field F(x, y)=(ycosx)i +(sin x+ y)j being conservative? Please show.

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Q: Show that F = (2xy + z³)ï + (x²)j+3xz²k is a conservative field. Find the scalar potential and the…

A: Given F = 2xy+z3 i ; x2j + 3xz2kThe objective is to show that given vector fieldis coonservative…

Q: (a) Suppose a vector field F(x, y) construct a potential function o(x, y). i+ is conservative. Then

A: Consider the given vector field.

Q: Find the potential function p, for the conservative vector field F(x, y, z) = (2xe² – y²sin x)î +…

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Q: Determine the line integral if the vector field is F(x, y) = x²yî – 2xyf On a rectilinear path that…

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Q: Show that the field F = (cos x)i + (- y sin y)j + (z3)k is conservative and Önd a potential function…

A: F = ( Cosx )i + ( -y Siny )j + ( z3 )k  F = gradient ( - Sinx - y Siny - Cosy + z4/4 ). Thus, we can…

Q: Briefly describe how to find a potential function φ for a conservativevector field F = ⟨ƒ, g⟩ .

A: Before we get into the procedure to find the potential function, we need to understand certain…

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 conservative vector field for the potential function by finding its gradient. f(x, у, 2)…

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Q: Consider the line integral cos(x+2y)dx+2cos(x+2y)dy where S is a curve starting at (0,0) and ending…

A: We have to show that F is conservative. When F is given as F=cosx+2yi+2cosx+2yj For the conservative…

Q: Find the gradient field F = Vo for the potential function o = x²y-3y²x. F =

A: Given the potential function:φ=x2y-3y2xWe need to find the gradient field F=∇φ

Q: Show that the isoclines of dy/dt= t are vertical lines. Sketch the slope field for −2 ≤ t ≤ 2, −2 ≤…

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Q: Explain why a potential function for a conservative vector field is determined up to an additive…

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Q: 5. Suppose that the vector field F = is conservative. Find a potential function of the vector field…

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Q: (xy³ +3x²y, 3xy² + x²) 2+x ху у,

A: According to question given that Vector Field F⟨xy3+3x2y, 3xy2+x2⟩

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: Find a potential function for the conservative vector field. F(x, y) = xex*4Y(2yi + xj)

A: If curl f is zero then conservative field

Q: Show that the vector field F = (yz² ye, rz² - e, 2xyz) is conservative, and find a potential…

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Q: Find a potential functioon for the conservative vector field given by y3/2 3 xy + z, 2 sin z 2.x…

A: According to question,Given vector field,F→x, y, z = 2xx2 + cos z-y322x, -32xy+z, -sin zx2 + cos z+y

Q: 6. Use the fundamental theorem of line integrals to evaluate 2x dr + 2y dy where C is the line…

A: Let's understand the Fundamental Theorem of Line Integrals:

Q: Is the area F = 16xi+ (2 − y2) j conserved? If the conserved field is, find the potential function.

A: Here we find the potential function by integration.

Q: Find the line integral F.dr of the vector field F=ax] over a curve C defined by a C rectangle in the…

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Q: Show that the vector field =(-2 x²y xy²¹ is conservative by finding all its potential functions.…

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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: 2. Find the gradient field F = Vy for the potential function p. Sketch a few level curves of o and a…

A: Given φ(x,y)=2x2+4y2 The gradient field  F=∇·φ=∂φ∂x,∂φ∂y=4x,8y Some vectors of F are  At x=1 and y=1…

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: 3. Find a potential for the conservative vector field: F(x, y, z) = (2xy + 2x, x² – 2yz³, -6z –…

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Q: Express the field D = (x2 + y)-(xax + yay) in cylindrical components

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Q: Find the conservative vector field for the potential function by finding its gradient. g(r, y, :) +

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

A: We have to find out gradient

Q: Show that the vector field F = 2xyi +(x² - 2y)j is conservative and evaluate the line integral (F•dš…

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Q: 6. Show that F = [y? cos x + z3, 2y sin x – 4, 3xz2 + 2] is a conservative force field. Find the…

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Q: Find a vector potential field for Ġ = [6xz+x³, –3r²y – y², 4.x + 2zy – 32²], i.e. find F such that V…

A: The solution is below.

Q: Find the potential function f for the field F = 6xi + 6yj + 4zk. %3D

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Q: Calculate the line integral in scalar field ∫C (x2y2-√x)dy is the arc of the curve y=√x of (1,1) to…

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Q: Show that the line integral is independent of path. 2xe-Ydx + (2y - x2e-Y)dy, Cis any path from (1,…

A: The product rule of differentiation states that d dxfg=fdgdx+gdfdx, where f, g are functions of x.…

Q: The integral of the vector field F(x, y) = (cos |x|, sin y|) A = (-2,0) and final point B (0, 2),…

A: Correct answer is (B) sin(2) - cos(2) + 1 Explanation:

Q: • (2,1,1) 20. (2x In y – yz) dx + хz ) dy — ху аdz - у (1,2,1)

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Q: Consider a conservative vector field: F(x, y) = (y cosx + y2, sinx + 2xy – 2y) Find a potential…

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Q: Compute the divergence and curl of the vector field cos (22 ) sin² (z² yz) 10 yz4 – xyz2 at the…

A: We have to find divergence and curl of vector field:

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

A: Solution