Verify Green's Theorem for the vector field F = (y², sin x) and the domain contained inside the closed triangular path, with vertices (0, 0), (2, 1)&(1, 2), oriented counter-clockwise. Note, you may use a change of variables on the double integral side.

Q: Let F(x, y, z) = yi + xzj + 2xyk and G(x, y, z) = zi + xj + yk. Refine expression (FV)G (F₁0₁ + F202…

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Q: For the vector field G = (yey + 3 cos (3x + y))i + (xeªy + cos(3x + y)), find the line integral of G…

A: We find the required integrtion using fundamental theorem of line Integral.

Q: Compute the vector assigned to the points P = (1, 1) and Q = (-3, -3) by the vector field F = (-y,…

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Q: Compute the flux of the vector field F = yi + 3j – xzk through the surface S, which is the surface y…

A: We compute the flux of the vector field through the surface.

Q: The figure shows the vector field F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end…

A: Given, vector field F→x,y=2xy,x2 and three curves that start at 1,2 and end at 3,2.

Q: Sketch and describe the vector field F (x, y) = (-y,2x)

A: The given vector field is F→x,y=-y,2x A vector field F→ in ℝ2 or ℝ3 is an assignment of a…

Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

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Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: Consider the vectorial V=2+ŷ+ 2 z . . function=z²+x²y + y²2 and the gradient operator Please…

A: Given that the vectorial function ψ→=z2 x^+x2 y^+y2 z^ The gradient operator ∇→=x^.∂∂x+y^.∂∂y+z^.∂∂z…

Q: Evaluate the 1-form φ = x2yz dx − z dy on the vector field V (x, y, z) = xU1(x, y, z) − yzU2(x, y,…

A: We need to evaluate the 1-form of  ϕ=x2yz dx - z dy.....(1) in the vector field  V(x,y,z) = x…

Q: Find the flux of the vector field V(x, y, z) = 2xy²i+ 5x²yj + 3z³k out of the unit sphere. Flux =

A: Denote the unit sphere by S If the sphere is closed,  then the flux of v(x, y, z) out of S is given…

Q: Find a potential function of the vector field. F(x,y) = (3x²y + 1, x³+2y)

A: we have to find potential function.

Q: Let C₁ be the line segment from the point (-4, 8) to the point (2, -4), C₂ be the arc on the…

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Q: Find the parametrization of two different curves from the point (2,4) to (3,9). Compute the work…

A: Given point (2,4)  and (3,9)now equationof line passing through two point is given by…

Q: What is the geometrical meaning of an integral of a vector function?

A: We can realize geometric meaning of integration such as

Q: Calculate vector field F = (2-z²+2-x²)i + (2 · y³ + 4·x)j + (2x²z³+2)k a flux from a box with the…

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Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

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Q: Show that the vector field F(x, y, z) = (-ycos(8x), 8x sin(-y), 0) is not a gradient vector field by…

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Q: Question 3 Consider the vector field v(x) = i+ i+ 6k. y j + 6k. Vx²+y Vx²+y² a) Calculate /v• dr,…

A: Given vectors is vx=xx2+y2i+yx2+y2j+6k To evaluate ∫Cv·dr where C is the line joining (8,0,8) to…

Q: Show that Vx (V.f) = 0 for any vector field f.

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Q: Find the curl of the vector field u = a) b) ▼xu =( V. (V x u) = = (7xyz², 6x² y³, 2x³ y³ z4 ).

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Q: Let F(x, y) = ⟨xy^2+2x+1, x^2y − 3y⟩ be a vector field. Let C be the curve that consists of two line…

A: Given that  Fx,y = xy2+2x+1 , x2y - 3y is a vector field. Let C be the curve that consists of two…

Q: 5) Find the flux of F = (-2y, -x) on the circle of radius 2 centered at the origin. Use the vector…

A: When the field vectors are going the same direction as the vectors normal to the surface, the flux…

Q: Sketch the vector field of F = (-y,0) in the xy-plane. %3D

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Q: Let f (x, y) be a harmonic function such that ∇f (1, 1) = (−1, −3). Show that the point P (1, 1) is…

A: To show that P(1,1) is a sink for the vector field G, we need to show that the vectors in the…

Q: Find the curl of the vector field F =< yx5, xz², zyб curl F = i + 1:0 12

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