Find a constant a such that the vector fieldis conservative.a =F(x, y) = 2 x· · (3 · y + 1)i + a · x²jNext find a scalar potential for the vector field F such that (0,0) = 3.44(x, y) =Vector field F

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Answered: is conservative. a= F(x, y) = 2.x (3.y…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Consider the vector fieldF(x, y) = (Ax3y + 4xy + By2)i + (x4 + 2x2 + 8xy)j (a) Find the values of A and B which make F a conservative vector field on ℝ2. Enter the values of A and B separated by a comma. (b) For the values of A and B found in part (a), find a potential function f (x, y) for F that satisfies f (0, 0) = 0. (c) If F(x, y) = ⟨P(x, y), Q(x, y)⟩ is the vector field found in (a), compute the line integral ∫ C P(x, y) dx + Q(x, y) dy,where C is any path starting at (0, 0) and ending at (1, 1).
Show that the vector field F(x, y) = ( 2xy, x² +2y) is conservative. Then, find the potential function and -(4,2) use it to evaluate F.dr (1,0)
Cal 3
Calculate vector field F = (2 · z² + 2 · x²)i + (2 · y³ + 4 · x)j + (2 · x² · z³ + 2)k . a flux from a box with the coordinates of opposite corners (1,0,0) and (2,1,1) and you will be parallel to the coordinate planes. Use Gaussian divergence theorem fF.dS = SSS V · FdV. PM
b) Verify whether the vector field F is conservative, if so, then find the potential function F = (E,- 1). z -xz x f(x, y,z). The vector field is y2 'y
An airplane took off from point (3, 0, 0) at an initial velocity, v(0)= 3j with an acceleration vector of a(t) = (-3 cos t)i + (-3 sin t)j + 2k. Determine the vector function, r(t), to represent the path of the airplane as a function of t.

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is conservative. a= F(x, y) = 2.x (3.y + 1)i + a.x²j Next find a scalar potential for the vector field F such that (0,0) = 3. 4(x, y) = Vector field F