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v(x. y)= (ar² + 3y². yy), (x,y) € R², where a, 3, 7 are nonzero real numbers. (i) Let C be a circle of radius R> 0 centered at (a,b) R2 such that the distance from (a, b) to (0,0) is greater than R. Compute the index of e around C. (ii) Let C be any closed curve in R2 not enclosing (0,0) inside. Compute the index of e around C. (iii) In either case (i) or (ii), show that v(x, y) = 0 indeed has no solution inside C and explain your finding in view of your results in (i) and (ii). (Hint: It may be hard to compute (i) directly. However, you may study what happens when you shrink R down to zero. You need to justify why you can do so.)

v(x. y)= (ar² + 3y². yy), (x,y) € R², where a, 3, 7 are nonzero real numbers. (i) Let C be a circle of radius R> 0 centered at (a,b) R2 such that the distance from (a, b) to (0,0) is greater than R. Compute the index of e around C. (ii) Let C be any closed curve in R2 not enclosing (0,0) inside. Compute the index of e around C. (iii) In either case (i) or (ii), show that v(x, y) = 0 indeed has no solution inside C and explain your finding in view of your results in (i) and (ii). (Hint: It may be hard to compute (i) directly. However, you may study what happens when you shrink R down to zero. You need to justify why you can do so.)

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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v(x, y) = (ar² + 3y², yy), (x,y) = R², where a, 5, 7 are nonzero real numbers. (i) Let C be a circle of radius R> 0 centered at (a,b) € R2 such that the distance from (a, b) to (0,0) is greater than R. Compute the index of v around C. (ii) Let C be any closed curve in R² not enclosing (0,0) inside. Compute the index of v around C. (iii) In either case (i) or (ii), show that v(x, y) = 0 indeed has no solution inside C and explain your finding in view of your results in (i) and (ii). (Hint: It may be hard to compute (i) directly. However, you may study what happens when you shrink R down to zero. You need to justify why you can do so.)
Transcribed Image Text:v(x, y) = (ar² + 3y², yy), (x,y) = R², where a, 5, 7 are nonzero real numbers. (i) Let C be a circle of radius R> 0 centered at (a,b) € R2 such that the distance from (a, b) to (0,0) is greater than R. Compute the index of v around C. (ii) Let C be any closed curve in R² not enclosing (0,0) inside. Compute the index of v around C. (iii) In either case (i) or (ii), show that v(x, y) = 0 indeed has no solution inside C and explain your finding in view of your results in (i) and (ii). (Hint: It may be hard to compute (i) directly. However, you may study what happens when you shrink R down to zero. You need to justify why you can do so.)
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