6. Consider the vector field F = (-y,x). a. Check if the vector field is conservative. b. Consider the line integral SF dr, where C is the counterclockwise (positively oriented circle of radius 3 centered at the origin. i. Calculate the line integral using the regular line integral method, using either SF(t) · r'(t) dt or S.P dx + Q dy.

Q: Solve the line integral with the vector field F (x, y) = (xy, seny) and the parametrization x (t) =…

A: The solution is given as

Q: Consider the vector field F(x, y) = (3y² – 2xy, 6ry – x²). (a) Show that F is conservative. (b) Find…

A: According to question, Given vector field, F→x, y = 3y2 -2xy, 6xy-x2

Q: Consider the vector field F (2x + 5y, 5x + 7y) Is this vector field Conservative? Conservative If…

A:

Q: (a) Given the vector field F = xy + 2yzý + 3zx 2, calculate the line integral around the closed path…

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Q: The following figure shows a half-cylindrical closed surface, defined by the cylinder x² + y² = 4…

A: As per the question we have to find the total flux of the vector field V through the flat top…

Q: Consider the vector field F = (5x + 6y, 6x + 4y). F - dr along the curve C given by r (t) = (t², t*)…

A: ∫CF→·dr→=∫abF(r(t)) r'(t) dr

Q: 2. Find the curl of the vector field. F = z°e* sin(2 y)i – e' cos yj+ xyzk to

A:

Q: If F (x, y) = x2 i + y2 j is a conservative vector field, find its potential function. With the…

A: The solution are next step is

Q: Find the line integral of the field F = over the portion of the ellipse * + y = 1 lying in the first…

A:

Q: Given a scalar field f(x, y)= xy + x² and C is the curve defined by the equation y =-x². Calculate F…

A: Given scalar field  fx,y=xy+x2 and C is the curve defined by the equation y=-x2.

Q: Find the flow of the field F = V(x²ze) once around the ellipse C in which the plane x + y + z = 1…

A: Consider the given field, F=∇x2zey around the ellipse in which the plane x+y+z=1intersects the…

Q: Given the following vector field and orientated curve C, evaluate F•Tds. (x,y) F = on the curve r(t)…

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Q: Find the work done by the vector field F = (– 4z, – 3y, – 2x) in moving an object along C in the…

A:

Q: Evaluate the line integral F. dr, with vector field F(x, y, z) = sin xi + cos yj + xz k, and path C…

A:

Q: 2) Given the vector field F = (- Vx²+y² 2z2) write the integral to calculate the outward x²+y² flux…

A: We have to find the outward flux across the cylinder x2+y2=100.

Q: Consider the vector-field F(x, y) = y²i-xj, and let C be the arc of the parabola y = k – x² from the…

A:

Q: Given a vector field F = (x + xy?)i + (5xy)j. Use Green's Theorem to evaluate F- di where C is a…

A:

Q: Consider the following vector field F (x, y, z) = yz cos (xy) i+ xz cos (xy) j+ sin (xy)k (a) Use…

A:

Q: Let W be the work done by the forced field F(x,y,z) =yi+zj in order to move a particle along the…

A: The formula for work done is W=∫F.dr.

Q: Given the conservative vector field, F=(x +y³)i +(2xy +5)j , evaluate th integral F•ds where C is…

A: We evaluate the given integral by using line integral. We parameterized the curve between two…

Q: Consider the vector field F = (x² + y², 7xy). Compute the line integrals f. F· dr and f F· dr, where…

A: Given F=x2+y2,7xy, c1t=t,t2 and c2t=t,t for 0≤t≤1. We know that ∫cF·dr=∫abFxt,yt·C'(t)dt. When…

Q: Find the counterclockwise circulation and outward flux of the field F = xyi + y2j around and over…

A: Theorem used:

Q: Show that the vector field given below is conservative and find its potential function. F(x, y) =…

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Q: Consider I = ∫CF⋅dr, where (img17) is a conservative vector field and curve C is parameterized by:…

A:

Q: For vector field F(x, y, z)=(1+ 9xry, 3x +e, ye' +2= (a) Carefully calculate curl F. (b) Find the…

A: We have to calculate curlF  and total work

Q: Consider the vector field F and the curve C below. F(x, y) = (3 + 4xy2)i + 4x²yj, C is the arc of…

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Q: Consider the vector field F = 2xy i + (x2 + 2y) j. Verify the Green's Theorem in the closed region…

A:

Q: Given the following vector field and oriented curve C, evaluate T ds. F = (x.y) on the parabola r(1)…

A: In this question we will find the line integral of F over c.

Q: Given the following vector field and oriented curve C, evaluate |F•T ds. F= (x,y) on the parabola…

A:

Q: For what values of a and d does the vector field F = ⟨ax, dy⟩ have zero flux across the unit circle…

A: A vector field, have zero flux. Across the unit-circle centered at the origin and oriented…

Q: Use the divergence of the vector field to compute F(x, y) = (tan-1 (2²)) i + (xy-2 +82²) j 4 fo…

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Q: Compute the line integral of the vector field F = (4y, –4r) over the circle z2 + y? = 16 oriented…

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Q: Consider the vector field F=(x^2+y^2,10xy). Compute the line integrals ∫c_1 F⋅dr and ∫c_2 F⋅dr,…

A: Given:

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

A:

Q: F=cos(x) î+sin(y) ĵ+cos (z) k

A:

Q: Given the following vector field and oriented curve C, evaluate F•Tds. C F= (x,y) on the parabola…

A: Given,F = <x,y>and r(t) = <14t, 3t2> , for 0≤t≤1. now,

Q: Compute the line integral of the vector field F = (4y, –4x) over the circle x² + y? = 36 oriented…

A: Given: F= <4y,-4x>          x2+y2=36

Q: 16. Compute the flux V· nds of the vector field V(r, y) = (x² – 2ry, y² – 2xy) across the curve C :…

A: We have to compute the flux ∫ℂV·nds of the vector field Vx,y=x2-2xy,y2-2xy across the curve…

Q: Let D be the upper half-plane (y > 0) and consider the vector field F : D+R² given by F(x, y) = (…

A:

Q: Consider the vector field F and the curve C below. F(x, y) = x2y³i + x³y?j, C: r(t) = (t2 – 2t, t2 +…

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Q: Find the flux of the field F = z k across the portion of the sphere x2 + y2 + z2 = a2 in the first…

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Q: Why does a two-dimensional vector field with zero divergence on a region have zero outward flux…

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Q: Evaluate the line integral of the vector field F = xy²i+ x²yj along the curve r(t) = (t+sin,t+cos-…

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Q: Consider the following vector field. F(x, y, z) = (2e* sin(y), 7e sin(z), 4e² sin(x) (a) Find the…

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Q: Consider the vector Field 7 and the curve C below. F(x,y) = (3 +4 ху?) 1 + 4x²yј, C is the arc of…

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Q: Evaluate F Î ds for the vector field F(x, y) = yì – xj counterclockwise along the circle x² + y² = 1…

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Q: Given vector field F(x, y) = (4x³y? – 2xy³,2x*y – 3x?y² +6y³) and curve C defined by C: 7(t) = (t+…

A: Here we will find out the required value.

Q: Given the following vector field and oriented curve C, evaluate JF•T ds. F= (xy) on the parabola…

A:

Q: Consider the vector field F and the curve C below. F(x, y) = (4 + 2xy²)i + 2x²yj, C is the arc of…

A: