You are given the vector field F(x, y, z) = [3 · x · sin(y · z), cos(2 ⋅ x) ·∙ z, −x · y]. . 1) Compute its curl. Enter your answer as a list of components. For example, if the curl has components 1, 2 and 3 enter [1,2,3]. VXF= [?,?,?]

Q: Find the curl of the following vector field: F = (z, 2y, -3x)

A: Follow the procedure given below.

Q: 33. Define T: R³ → R³ by y + 2z 2x +3y - z x +2y - 2z x - y a. Find all vectors in R' that are…

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Q: 1. Match each vector equation with its graph. i. r(t)=ti-tj+√√2-1²k iii. r(t) = sin(t) i + cos(t)j +…

A: Given:(i) (ii) (iii) .(iv) .To Match: each vector equation to its sketch:

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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Q: Find the unit tangent vector T(t) for r(t) = 2ti + 3tj + 6tk at t = −2. ○-4i-6j-12k ○_1 O 3 :- i- 49…

A: To find the unit tangent vector T(t), we first need to find the derivative of r(t): r(t) = 2ti + 3tj…

Q: Prove the following vector identity using index notation: A × (B × C) = (A · C)B — (A · B)C.

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Q: 2. Match each of the following vector fields with its respective plot by filling in the blanks. (x²…

A: Given: Vector fields 1) x2-y2-4, 2xy 2) y2, x2 3) x1-y, y(x-1) 4) x, 3y

Q: Question 1 Define the two vector valued functions of one variable u(t)= (-1/t, 2t, t²), v(t) := (t²,…

A: Let's find derivative of vector function.

Q: Prove the following vector identity using index notation: (A. B)² + (A x B) · (A × B) = |A|² |B|².

A: We need to prove the following vector identity, A·B2 + A×B·A×B=A2 B2 Consider right hand side,…

Q: dr = wa x b dt r X

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Q: 5. Let A E Mmn(F). Prove that v E F™ is in colspace(A) if and only if there exists some vector u e…

A: Here, we assume the matrix as linear transformation.

Q: A vector field in polar coordinates has the form F(r.0) = f(r.0)u, +g(r,0)ug, where the unit vectors…

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Q: d) Verify the Stoke's theorem for vector field E = xyx – (x² + 2y²)ỹ by evaluating the following: i.…

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