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1 : Evaluate f (x + 3y)dx + ydy where C is the Jordan curve given by the graphs of y = e, y = e- and the horizontal line y = e-1 a) By Green's theorem b) By direct computation.

1 : Evaluate f (x + 3y)dx + ydy where C is the Jordan curve given by the graphs of y = e, y = e- and the horizontal line y = e-1 a) By Green's theorem b) By direct computation.

Advanced Engineering Mathematics
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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Question no 1

1: Evaluate (x+3y)dx + ydy where C is the Jordan curve given by the graphs of y = e, y = e- and the horizontal line y = e-¹ a) By Green's theorem b) By direct computation. 2: Let F = (2x cos(x² + y²), 2 + 2y cos(x² + y2)) and C be the curve joining (-1,-1) to (-1,0) by traveling on the curves y = ³ from (-1,-1) to (0,0), then 1 y = r² from (0,0) to (1,1) and finally on y = (x + 1)(x-3) from (1,1) to (-1,0). Evaluate ·[F F. dr. 3: A function f(x, y, z) is called homogeneous of degree d if for any real number AER, f(xx, Ay, Az) = f(x, y, z). Assume that F = Vf is a conservative vector field where f(x, y, z) is a homogeneous function of degree 4. Evaluate (1, 1,-2) to the point B = [F F dr where C is a curve joining point A = 1 (-2,-2,4), and f(1, 1, -2) 32 vector field F = curve C on R². = 4 Let f(x) be a polynomial function of degree greater than 3 and define a (yf' (a), f(x)). Show that fr F. dr 0 for any simple closed
Transcribed Image Text:1: Evaluate (x+3y)dx + ydy where C is the Jordan curve given by the graphs of y = e, y = e- and the horizontal line y = e-¹ a) By Green's theorem b) By direct computation. 2: Let F = (2x cos(x² + y²), 2 + 2y cos(x² + y2)) and C be the curve joining (-1,-1) to (-1,0) by traveling on the curves y = ³ from (-1,-1) to (0,0), then 1 y = r² from (0,0) to (1,1) and finally on y = (x + 1)(x-3) from (1,1) to (-1,0). Evaluate ·[F F. dr. 3: A function f(x, y, z) is called homogeneous of degree d if for any real number AER, f(xx, Ay, Az) = f(x, y, z). Assume that F = Vf is a conservative vector field where f(x, y, z) is a homogeneous function of degree 4. Evaluate (1, 1,-2) to the point B = [F F dr where C is a curve joining point A = 1 (-2,-2,4), and f(1, 1, -2) 32 vector field F = curve C on R². = 4 Let f(x) be a polynomial function of degree greater than 3 and define a (yf' (a), f(x)). Show that fr F. dr 0 for any simple closed
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