1) Consider the simple, piecewise, oriented, closed curve C shown in the figure and the non-conservative vector field given by F (x, y) (cos(x) + y?,In(y) – 6 y = -2x + 8 y = x2 1 2 3 4 When calculating F * dr is obtained as an approximate result: A) -10.1491 B) 10.1491 C) -74.9333 D) 74.9333

Q: 5. Consider the vector field F(x,y,z) = yzi + xzj+(xy+2z) k and the curve C given by r(t) = (t+ sin…

A: Given vector field: Fx,y,z=yzi+xzj+xy+2zk Curve C is given by: rt=t+sin12πt,t+cos12πt,2t,   0≤t≤1…

Q: Verify Green's theorem F - dř = P(r, u) dzr + Q(z, y) dy = | | {0.Q(z, y) – ,P(x, y)} d.ædy, %3D for…

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Q: 4. Consider the vector field given in Cartesian coordinates by A(r) = (x² + y³, 3xy², z²). (a)…

A: Given : Ar=x2+y2,3xy2,z2

Q: 14) Consider the closed, simple, piecewise smooth, oriented, simple, piecewise smooth closed curve C…

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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: 4. consider F(x, y) = (y² − 2xy) i + (2xy − x²) j. (a) Show that F is a conservative vector field in…

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Q: 5. Let F(x, y) = ei + xej be defined on a smooth curve C given by the vec function r = ti+t2j for 0…

A: A detailed solution is given below

Q: Let the vector field F (x, y) = (3x2y-2+ 2xy-1)i + (-2x3y-3 − x2y-2)j, be a conservative field.…

A: We will use Integration and concept of total differentiation for finding scalar potential of vector…

Q: 15) Consider the vector field F (x, y, 2) = (In y, x, z) and the oriented curve C:r (t) = (t, e', 2…

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Q: Show that the following vector field is conservative by using a computer. Calculate F. dr for the…

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Q: 8. The component of the field F= (1,2,3) normal to the plane z = -3x + 3y -1 is (a) 0. (b) –2//19.…

A: If we are given a plane equation in the form Ax+By+Cz=D, then the normal vector to the plane will…

Q: Evaluate both sides of the divergence theorem for the vector field D = 2xyx + x²9 [C/m²] and a…

A: This question is belong to the vector calculus in which we have to deal with the divergence theorem…

Q: 7. Determine if the given vector field F is conservative on some domain and, if it is, find the…

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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: 3. Consider the vector field F = (y, -x, x² + y?). Find the flux of F through S if: 9– x2 – y? above…

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Q: 1) Consider the conservative vector field given by: F(x, y) = (exy3 + 2e2xy, e2x + 3exy2) A…

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Q: 2.) Calculate the curve integral over the vector field V(17, y) = ( * -2²y xy? for the positive…

A: Disclaimer: Since you have asked for a handwritten solution and currently we are not providing any…

Q: 3. Consider the vector field F = 2xy i + (x² + 2y) j. Verify the Green's Theorem in the closed…

A: Definition of green's theorem :

Q: 8. Show that the vector field, -y a² + y?' x² + yj²' ) is not conservative in R³, despite the fact…

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Q: 10. Evaluate the divergence of the following vector field at the point (0,1,2): F = (xyz, x? + y? +…

A: Given: F=xyz,x2+y2+z2, xy+yz+xz       The point 0,1,2 To find the divergence

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

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Q: 14 4. Consider the vector field F and the curve c below. 5 5 F(x, y) = x²y³ ₁ + x³y² Ĵ. C:r(t)…

A: The given problem is to evaluate the given path integral of the vector field over given curve.

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

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Q: Calculate the flux of the field F=xi+yj across the closed plane curve C, where C is the closed…

A: Given a field: F=x2i+y2j We need to find the flux of this field across a closed plane curve C which…

Q: Consider the surface integral of the vector field F = [xy², 3x², x²z] over a surface S, where S…

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Q: 2. Given that F(x, y, z) = (e, xey + e², ye²) is a conservative vector field. a. Find a potential…

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

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

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Q: (c) Determine the flux of the vector field F = x³i – 2yĵ + 7z²k across a cone z = x2 + y2 + 3 that…

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Q: 5. Show that F =(y cos(x),sin(x)+ y) is a conservative vector field and use this to calculate the…

A: Given: F → = y cos(x), sin(x) + y. A vector field F→ = P, Q is said to be conservative if partial…

Q: Is the vector field F(x, y) = (6ry cos(3x²y + e*) +e* cos(3x?y + e*), 3x² cos(3x²y + e*))…

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Q: 3. Consider the vector field F(x, y) = (2xy, x² – sin y). Evaluate %3D F. ds where C is the curve…

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Q: Ex: Find the work done by the vector field F = (y-x²)i + (z - y²)j + (x-z²2)k Over the curve r(t) =…

A: Given: F=y-x2i+z-y2j+x-z2krt=ti+t2j+t3k

Q: 1 Find the divergence of the vector field F -(xi + yj+ zk) Vx? +y? +Z 2 x² + y² + z? 1 x² + y? + z?…

A: If the vector field is given by F=F1i+F2j+F3k then the divergence of F is ∇·F=∂F1∂x+∂F2∂y+∂F3∂z.

Q: Consider the following vector field: F = (y* + 1, 3ry + 1). (8.1) Is ſ. F · dr path-independent?…

A: The given vector field is F=y3+1, 3xy2+1. (8.1) For a vector field F=P,Q, the line integral ∫CF·dr…

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

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Q: 1: Let Let F = (yz², tan-1(xy), 1 + xy³). %3D a) Show that F is not conservative. b) Show that F…

A: conservative vector field is a vector field that is the gradient of function

Q: 1).Evaluate ∮CF•dr, whereF=⟨e^x−y^3,cosy+x^3⟩, and C is a circle of radius 2, centered at the…

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Q: 2.4) Use the divergence theorem to find the outward flux of the vector field F(x, y, x) = x³i+ y³j+…

A: The divergence of the given vector field is  ∇·F=∂∂xx3+∂∂yy3+∂∂zz3=3x2+3y2+3z2 Let, S be the surface…

Q: 2. Show that F is a conservative vector field and then use that fact to evaluate SF. dr along the…

A: We need to show F is conservative vector field.

Q: 2 1 Given the vector field F=rcos a.+ =sing a tra evaluate the curl of the vector field.

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Q: Calculate the flux of the field F=xi+yj across the closed plane curve C, where C is the closed…

A: theory: graph:

Q: 9. (~Ch16 Review #13) Find the work done by the conservative vector field F(x, y) = along the curve…

A: given a conservative vector field  Fx,y=4x3y2-2xy3,2x4y-3x2y2+4y3 and a curve  rt=t+sinπt,2t+cosπt…

Q: 7. Let S be the surface of a solid cube with domain D = {1 < z < 2, 1 < y < 2,1 < z< 2}. For field F…

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Q: F(x, y, 2) = (y² cos r + 2*)i+ (2y sin æ – 4)j+ (3xz² + 2)k Cos z

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Q: 4- Evaluate the surface integral of the vector field F = = [ax, by, cz] over the sphere a2 + y² + z²…

A: From the given problem:   F→=axi+byj+czk   Sphere:  x2+y2+z2=36.   From the divergence theorem:…

Q: 40. Find the circulation of the field F = yi + (x + 2y)j around each of the following closed paths.…

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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: Show that the vector field F(x, y)= (12x² +3y² +5y)i+(6xy– 3y² +5.x)j is a conservative vector field…

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Q: Let the curve C be the boundary of the triangle cut from the plane 2x +y +3z =3 by the first octant,…

A: We will use Stokes Theorem to find the circulation: ∮CF⋅dr=∬S(∇×F)⋅n^dS

Q: B) Find the work done by the conservative vector field F (x, y, z) = yzi+xzj+xyk along a smooth…

A: Since, it is given that the field is conservative.