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Using Green's theorem, evaluate Fir)-drcounterclockwise around the boundary curve C of the region R, where 1. F=Bry. ). R the rectangle with vertices (0, 0). (3, 0). (3, 2). (0, 2)

Using Green's theorem, evaluate Fir)-drcounterclockwise around the boundary curve C of the region R, where 1. F=Bry. ). R the rectangle with vertices (0, 0). (3, 0). (3, 2). (0, 2)

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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PROBLEM SET 10.4 1-12 EVALUATION OF LINE INTEGRALS BY GREEN'S THEOREM 13-16 INTEGRAL OF THE NORMAL DERIVATIVE Using (9), evaluate ds counterclockwise over the Using Green's theorem, evaluate Fir) dr counterclockwise around the boundary curve C of the region R, where boundary curve C of the region R. 13. w sinh x, R the triangle with vertices (0, 0). (2, 0), (2, 1) 14. w =x + y. C: + y = 1. Confirm the answer by direct integration. 15. w 2 In (x + y+xy. R: 1Sys 5 -.x20 16. w = xy + xy"., R: x +ys 4. y2 0 1. F-y . R the rectangle with vertices (0, 0). (3, 0). (3. 2). (0. 2) 2. F = [y sin x. 2r cos y). R the square with vertices (0,0%. (.0), m 까 (0. m 3. F [-y. , Cthe circle x+ y 25 4. F = [-e. . R the triangle with vertices (0, 0). (2. 0). (2. 1) 5. F le 1 R the triangle with vertices (0, 0). (1. 1), (1, 2) 6. F [x cosh y. sinh y). R: xS y S x. Sketch R. 7. F = [? + y. - y). R: 1sys2 -. Sketch COs 17. CAS EXPERIMENT. Apply (4) to figures of your choice whose area can also be obtained by another method and compare the results. 18. (Laplace's equation) Show that for a solution w(x, y) of Laplace's equation Vw = 0 in a region R with boundary curve C and outer unit normal vector n, R. 8. F [e cos y.-e sin yl. R the semidisk dx dy (10) 9. F = grad (r cos cay), R the region in Prob. 7 10. F= [r In y. ye R the rectangle with vertices (0, 1). (3, 1). (3, 2). (0. 2) 11. F= [2r- 3y. x+ 5y), R: 16+ 25ys 400, y 20 12. F = -th), R: 1s+ s 4, x2 0, y 2x. Sketch R. ds. %3D 19. Show that w = 2e cos y satisfies Laplace's equation w = 0 and, using (10), integrate w(awlon) counterclockwise around the boundary curve C of the square 0 SxS 2,0 sys2.
Transcribed Image Text:PROBLEM SET 10.4 1-12 EVALUATION OF LINE INTEGRALS BY GREEN'S THEOREM 13-16 INTEGRAL OF THE NORMAL DERIVATIVE Using (9), evaluate ds counterclockwise over the Using Green's theorem, evaluate Fir) dr counterclockwise around the boundary curve C of the region R, where boundary curve C of the region R. 13. w sinh x, R the triangle with vertices (0, 0). (2, 0), (2, 1) 14. w =x + y. C: + y = 1. Confirm the answer by direct integration. 15. w 2 In (x + y+xy. R: 1Sys 5 -.x20 16. w = xy + xy"., R: x +ys 4. y2 0 1. F-y . R the rectangle with vertices (0, 0). (3, 0). (3. 2). (0. 2) 2. F = [y sin x. 2r cos y). R the square with vertices (0,0%. (.0), m 까 (0. m 3. F [-y. , Cthe circle x+ y 25 4. F = [-e. . R the triangle with vertices (0, 0). (2. 0). (2. 1) 5. F le 1 R the triangle with vertices (0, 0). (1. 1), (1, 2) 6. F [x cosh y. sinh y). R: xS y S x. Sketch R. 7. F = [? + y. - y). R: 1sys2 -. Sketch COs 17. CAS EXPERIMENT. Apply (4) to figures of your choice whose area can also be obtained by another method and compare the results. 18. (Laplace's equation) Show that for a solution w(x, y) of Laplace's equation Vw = 0 in a region R with boundary curve C and outer unit normal vector n, R. 8. F [e cos y.-e sin yl. R the semidisk dx dy (10) 9. F = grad (r cos cay), R the region in Prob. 7 10. F= [r In y. ye R the rectangle with vertices (0, 1). (3, 1). (3, 2). (0. 2) 11. F= [2r- 3y. x+ 5y), R: 16+ 25ys 400, y 20 12. F = -th), R: 1s+ s 4, x2 0, y 2x. Sketch R. ds. %3D 19. Show that w = 2e cos y satisfies Laplace's equation w = 0 and, using (10), integrate w(awlon) counterclockwise around the boundary curve C of the square 0 SxS 2,0 sys2.
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