3. Let F = (2y² + z)i + 4.ryj + rk. (a) Show whether F is conservative. (b) Using the fundamental theorem of calculus for line integrals, prove that F. dr = -T along a curve C' defined by a vector function r(t) = cos ti + sin tj + tk for (c) Using the definition, show that the curve C in part (b) is not closed.

Q: Let  f(x, y) = sin(15x2 + 3y2) Calculate the quadratic approximation of f at the point (0,0)

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A: Disclaimer: There are two questions here and as per Bartleby policy, I am solving only the first…

Q: [[(2x + y²) da + 2xy dy], where C is the line segment from (-1,0) to (1, 2) Evaluate in two…

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Q: 6) What is the tangent plane of f (x, y) = 5x² – 2xy? + 3xy + 4x at point (xo, Yo) = (1, –2) ?

A: The question is solved in step. 2

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A: Given:          r(t)=cost,sint,et at t=0           r→(0)=cos(0),sin(0),e0          r→(0)=1,0,1…

Q: Show that the parametric curve r(t)= , t [-1,1] is smooth and compute its lengh

A: The given parametric curve is: r(t)=t1+t2, 11+t2 ;          t∈-1, 1

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Q: 2. Consider the vector-valued function R(t) = Show that is continuous at t = 1. (e¹-², 1-1², (2-1)),…

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Q: S(x,y) =-x' +2x*y-² -4y+12 ED CALC

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Q: Let F (8xy, 3y, 8z). = The curl of F = (000). Is there a function f such that F = V f? ☐ (y/ (y/n)

A: Step 1: Step 2: Step 3: Step 4:

Q: prove that t1 = T2 is a necessary and sufficient condition for the unit function Id: (X, t1) → (X,…

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Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

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A: First we will plot the given parameters.

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

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A: To find - Solve for the area common with r = 2 + 5 cos2θ and r = 5sinθ

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