3. Use the divergence of the vector fieldto computeYF(x, y) 2²1) ₁ + (29-2 +²8²) J=fotan-1tan4-1(2x)dy -where C is the cardiod r= 1+ sin 0.(ry.Y2+82,dx,

Answered: Image /qna-images/answer/d10b5f00-d743-4d08-9fff-508e5dab60af.jpg

Answered: 3. Use the divergence of the vector…

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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6 Find the curl of the vector field F = < yx³, xz5, zy curl F= + S k
(7 T Show that e(t) = cos(t) + sin(t), – sin(t) + cos(t)) is the (a) Let F(r, y) = flow line of the vector field. (b) Let F(r, y, 2) = ( , e", 2²) be a vector field. Compute divF and curlF
Use the equation giving the flux of the vector field across the curve to calculate the flux of x + 1 y lã (x + 1)² + y²' (x + 1)² + y² F(x, y) = across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) I F. dr =
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.
Use the divergence of the vector field to compute F(x,0) = (tan 1(2x))i + (x3-2+8³) y) xy 4 fta (22) dy - (zy-2-54²) dr. tan¯¹ 4 xy dx, 2+8x² where C is the cardiod r = 1 + sin 0. j
Consider the vector field F(x,y) 2iu²j and the real-valued function f(x, y) = tan¬1(÷). One has V f(r,y) af it j dy (A) .... dx (-) (B) 1+(÷)² 1+ Since (-) i+ (C) Y 1+() 1+ly x2 + y? x2 + it follows that Vƒ = F. Hence F is conservative. Identify where the error(s) is (are) in the foregoing argument and give a brief explanation in words while perhaps also citing any relevant equation label(s) if appropriate in laying out your explanation.
Compute the divergence and curl of the vector field cos? (x2 ) F = sin (²y=) 2. 10 yz4 – ryz , at the point (1, 1, c). a. Divergence = ab sin (a) f Ω b. Curl = sin (a) f Ω a 00
Verify if the vector field F = (x² cos(3y), -³ sin(3y)) is conservative and evaluate the line integral Je where C' is the curve parameterized by R= for 0 < t < 1. cos(6) 3 A. I = B. I O c. I - OD. I = 0 E. I sin(6) 4 23 cos(3) 2 F. dR
Use the equation giving the flux of the vector field across the curve to calculate the flux of x+1 y (₁. (x + 1)² + y²' (x + 1)² + y² F(x, y): across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) [F C = Incorrect F. dr = 0.1074

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3. Use the divergence of the vector field to compute F(x, y) = (tan-1 (2x)) ₁ + (xy-2 +²8²) J j 4 fo tan-¹ (2x) 4 where is the cardiod r = = 1+ sin 0. dy - (zy - 2+8²) dr,