Consider the vector field F = (4yz, 4xz + 1, 4xy + 2z). O The vector field is not conservative OThe vector field is conservative, and the potential function such that f(0, 0, 0) = 0 for F is f(x, y) = If F is conservative, use f(x, y) to evaluate | F. dr along a piecewise smooth curve (C) from (0,3, -4) to (2,-5,5). F. dr

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Q: Consider the vector field v(x)

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

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A:

Q: 9. Sketch the vector field F(x, y) : :(yi — х). Vx? + y?

A: Consider the given filed, Fx,y=1x2+y2yi−xj Further simplify. Fx,y=yx2+y2i-xx2+y2i=〈yx2+y2,-xx2+y2〉

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Q: Let R be a differentiable vector-valued function such that R'(t) = (3t2, 2tvt2 + 1, 2t + 3) and R(1)…

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

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Q: 1. Let F =i+y^2j +z^4k. Find the general flow lines for this vector field. Present the flow line…

A: Consider the provided question,