6. Consider the vector field defined byyF = (²) ² + ( 3y² m (x) - ₂ √ √ ) ³ + ( √T)kij(a) Use the component test to show that the vector field F is conservative.(b) Find a potential function for F. Show the steps of your computation for this.A

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Answered: 6. Consider the vector field defined by…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose F = = (2y, -2x). Complete the following table of values of F. y = 1 y=0 y = -1 x = -1 B Values of F x = 0 x = 1 Using your table of values as a starting point, sketch this vector field on a piece of paper for -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2.
Consider the vector-valued function R (t) = ( 8/3 t 3− t, 2t2 , 2 − 2t2 ) - Find II R(t) II (Simplify the answer please) - Find the distance traveled by a particle moving along the curve from the point where t = −1 to the point where t = 2.
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6. Consider the vector field F(ar, y) = (8r°y+ e")î+ (Ka + e")3, where K is some constant. (a) (5 points) For which value of K is F the gradient of a function? For this value, find f such that F = Vf. (b) (5 points) For the value of K found in part (a), evaluate fe F dr, where C is the segment of the curve y = a from (0,0) to (1, 1). (c) (5 points) If instead K = 1, use Green's Theorem to evaluate f F dr, where C is the boundary of the triangle with vertices (0,0), (2,0), and (2, 1), traversed counterclockwise.
In the images below, the vector field F is shown by the blue arrows and the curve Cis shown in red. Indicate the value of F ·dT. 1. /, F ·dÌ = 0 2. ,F ·dT > 0 3. F dT < 0 4. The value of F ·dT cannot be determined with only this information. X
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6. Consider the vector field defined by y F = (²) ² + ( 3y² m (x) - ₂ √ √ ) ³ + ( √T)k i j (a) Use the component test to show that the vector field F is conservative. (b) Find a potential function for F. Show the steps of your computation for this. A