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(e) Give one example where the Green theorem fails, and explain how.

(e) Give one example where the Green theorem fails, and explain how.

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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Please help with (c)
Green's Theorem (a) State the Green theorem in the plane. (b) Express part (a) in vector notation. (c) Give one example where the Green theorem fails, and explain how. (d) Prove part (a) assuming that the domain D enclosed by the simple closed curve C with positive orientation is of the form D = {(x,y) | a ≤ y ≤ b, gi(y) ≤ x ≤92(y)}, where gi(y), 92(y) are continuous functions, and a, b are some real numbers. (e) Verify Green's theorem for a vector field F(x, y) = y²i+z²j and a triangle bounded by the lines a + y = 1 and -x + y = 1 and y = 0. (Hint: all the conditions in the Green theorem must be verified.)
Transcribed Image Text:Green's Theorem (a) State the Green theorem in the plane. (b) Express part (a) in vector notation. (c) Give one example where the Green theorem fails, and explain how. (d) Prove part (a) assuming that the domain D enclosed by the simple closed curve C with positive orientation is of the form D = {(x,y) | a ≤ y ≤ b, gi(y) ≤ x ≤92(y)}, where gi(y), 92(y) are continuous functions, and a, b are some real numbers. (e) Verify Green's theorem for a vector field F(x, y) = y²i+z²j and a triangle bounded by the lines a + y = 1 and -x + y = 1 and y = 0. (Hint: all the conditions in the Green theorem must be verified.)
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