For the vector field (x³z², o, -x²z), evaluate the line integral on the clase curve made by a square in the (x, 2) plane with corners at (1,0,1), (-1,01), (-1,0, -1), (1, 0,-1) in the counterclockwise direction.

Q: How do you graph the vector field F = ⟨ƒ(x, y), g(x, y)⟩?

A: Given: vector field

Q: 2. Sketch at least ten arrows in the vector field: F(x,y)=(-2y)i + (x+1)j

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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: Consider the vector field: F = (ex, In(xy), e xyz) (i) Find div(F) (ii) Find curl(F) (iii) Find…

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Q: Vz(x, y) V + y have the Show that all the vectors in vector field F same length, and state the…

A: The length or magnitude of a vector a, b is defined as follows: a, b=a2+b2 Note ca, b=ca, b, where c…

Q: c) Show that the vector field W(x, y, z) = (2yz+ xz – x², –2xz – yz + y?, y² – x² + z²) - - is a…

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Q: Find r'(t), r(t), and r'(to) for the given value of to r(t) = (1+t)i + ³j, to = 1 (No Response)…

A: We have to find the values of r't, rt0, r't0 where rt=1+ti^+t3j^, t0=1 and sketch the curve…

Q: Q4: Find the curl of the vector field F(x, y.2) = x*/yi +j+ zk at the point(2,4).

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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…

A: Given that v=<x-y, z+y+9, z2> and the given curve is y=1-x2-z2,y≥0 To find: The value of ∬S…

Q: Explain how to compute the curl of the vector field F = ⟨ƒ, g, h⟩.

A: Here given

Q: For the vector fields defined in part (b), evaluate separately each term in the iden- tity given in…

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Q: Consider the vector field v(x)

A: Given, the vector field        v(x)=yx2+y2i^-xx2+y2j^+6k^                               ...(1)

Q: Show that Gauss's Theorem holds for the vector field F(x, y)=(xy2, -yx2) in a side square 3.

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Q: Find the curl of the vector fleld F(z, y) = 9ye" i+2ejat the point (-2, 0). Enter an exact answer.…

A: F(x, y)=9y2 ex3 i +2x2 ey jcurl F→=∇×F→curl F→=ijk∂∂x∂∂y∂∂z9y2 ex32x2 ey0         =∂∂x(2x2…

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

A: Adding the plane surface y=0 to close the surface and apply the gauss divergence theorem.

Q: Find the flux of the vector field V(x, y, z) = 9xy²i + 9x²yj + z³k out of the unit sphere.

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Q: Compute the flux of the vector field (ï³, — xy³), out of the rectangle with vertices (0,0), (4,0),…

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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: Find r'(t), r(t), and r'(t) for the given value of to. Then sketch the space curve represented by…

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Q: 2. For the vector field F(x, y, z) = (x²y, xyz, −x²y²) F(x, y, z)=(x2y,xyz,-x2 y 2 ), find: (a) div…

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Q: Consider the vector field v(x)=(y/(x^2+y^2))i − (x/(x^2+y^2)) j + 4k. Calculate ∫Cv dr, where C is…

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Q: d) An open surface F is given by F = {(x,y, z) |z = 4 – x² – y² , z > 0} . Compute the flux of V…

A: V(x,y,z) = <2x+3y,2y+3x,-4z>F={ (x,y,z) : z=4-x2-y2 , z≥0}Consider, g(x,y,z) = z+x2+y2-4…