In (a)-(d), state whether the equation is an identity (valid for all F or f). If it is not, provide an example in which the equation does not hold.(a) curl (∇ f) = 0( c) div (curl (F)) = 0(b) div (∇ f) = 0(d) ∇ (div (F)) = 0

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

A: Given: vector field

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A:

Q: In words, what is a vector field? Give some physical examples of vector fields

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Q: When is the vector field continuous?

A: given,       Continuous vector field .

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A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

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Q: 1. Calculate the curl of F, V x F. 2. Calculate (V × F) · dS.

A: Given that the vector field F=(x2z, xyz, x2y)and the surfacez=1-x2-y2z≥0

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A: Given,  F→=yx4, xz5, zy3

Q: Problem 2. Let F be the vector field (xe² + cos(x), -ye², z sin(x)), and S the paraboloid 2 = 1- x2…

A: We will find out the required solution

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

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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〉

Q: State whether the curl of the vector field shown below is on average positive, negative or…

A: To Explain: whether the curlF is positive, negative, or zero in all given four different regions.

Q: Q3: Prove that the divergence of a curl is always zero.

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Q: Can you find a vector field whose curl is z?i? What about x2i? Explain your answer.

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Q: 14. Find the curl of the vector field F = x²i – xevj+ 2xyzk

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Q: The diagram shows a vector field F. What is the sign of V.F (P) cannot be ascertained positive zero…

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Q: Match the vector field with its graph. F(x, y) = yi -5+ -5+

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Q: Sketch the vector field. F(z, y) = (- 2 2 2 0 2 -2 0 -2 2 -2 2 -2 -2 2 -2 4- -4 -3 -2 -1 2 3 -2 -3-…

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Q: 5. Find the curl of the following vector field in both Cartesian and cylindrical coordinates. F=…

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Q: b) Find the curl of the vector field, CurlF, if F = Ax²z²i– By:'j+Cz*k, where the coefficients A,…

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Q: 4. Find the curl of the vector field F = %3D curl F = k

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Q: Find the curl of the vector field F = curl F - i.

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Q: Find curl(zi + xj+ yk).

A: Using definition of curl of vector.

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A: Given that:dydx=eay

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Q: Question Number 5: i) Find the divergence and the curl of the vector field F(x, y,z) = (xi + yj +…

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Q: 7. Find the outward flux of the field F(x, y) = -yi + x²j across the closed counter-clockwise curve…

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Q: Let F = (e", sin“ x, cos³ x). Calculate curl(F).

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