PROBLEM SET 10.41-12 EVALUATION OF LINE INTEGRALSBY GREEN'S THEOREM13-16INTEGRAL OF THE NORMAL DERIVATIVEUsing (9), evaluateds counterclockwise over theUsing Green's theorem, evaluate Fir) dr counterclockwisearound the boundary curve C of the region R, whereboundary 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 bydirect integration.15. w 2 In (x + y+xy. R: 1Sys 5 -.x2016. w = xy + xy"., R: x +ys 4. y2 01. 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. m3. F [-y. , Cthe circle x+ y 254. 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 -. SketchCOs17. CAS EXPERIMENT. Apply (4) to figures of yourchoice whose area can also be obtained by anothermethod 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 withboundary curve C and outer unit normal vector n,R.8. F [e cos y.-e sin yl. R the semidiskdx dy(10)9. F = grad (r cos cay), R the region in Prob. 710. 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 2012. F = -th), R: 1s+ s 4, x2 0,y 2x. Sketch R.ds.%3D19. Show that w = 2e cos y satisfies Laplace's equationw = 0 and, using (10), integrate w(awlon)counterclockwise around the boundary curve C of thesquare 0 SxS 2,0 sys2.

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Answered: Using Green's theorem, evaluate…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Let C be a piecewise curve consists of the following four line segments. First linesegment has vertices (0, 0) and (2, 0); the second segment has vertices (2, 0) and(2, 1); the third line segment has vertices (2, 1) and (0, 3); and the fourth linesegment has vertices (0, 3) and (0, 0). C is oriented counterclockwise. LetF(x, y) = ⟨x^2 − 2y, 2x⟩ be a vector field.(a) Sketch C accurately including the orientation.(b) Find the work done by F by moving a particle along the path C in the givendirection. Simplify your answer.
6) Apply Fundamental Theorem of Line Integrals integral to evaluate the line Shcxry)ds C ल where C is the square with vertices 10,0), (1,0), (1,4) and 10,1) traversed Counter clockwise and h (x,y) =( 3x²y+ xy%-1)i + lx²+x?y+ ky') f
3. (Section 17.4) Evaluate (a) Line integrals e for y dr + y dy where C is the circle z² + y² = 4 traced out once counterclockwise, using: (b) Green's Theorem
A,b,c
Use greens theorem to evaluate line interal (x2y)dx+y2dy),where c is the closed path formed be y=x and y=x3 from(0,0) to (1,1)
Let ∆ABC have vertices at A = (0,0), B = (3,4), and C = (12,5). Let ∆HFK have vertices at H = (6,0), F = (8,1.5), and K = (12,2.5). Is ∆ABC similar to ∆HFK ? Which similarity theorem would you use to check whether ∆ABC and ∆HFK are similar? Determine whether ∆ABC and ∆HFK are similar.
9. Let R be a square with vertices (0,0), (1,1), (2,0) and (1, -1) in the xy-plane. It might be useful/helpful to sketch the region R and the region S a) Find the image Sin the uv-plane under the transformation T: x=u + v, y = u - V hint: solve for u by solving for x + y (use system) b)Write the Jacobian Matrix of partial derivatives c) evaluate the determinant of the Jacobian d) Rewrite the integral using a change of variables to u and v with the Jacobian and evaluate the new integral. SSR xydA
Pls help ASAP
List the vertices in R² of the convex region defined by Ax ≤ b and x ≥ A = 1 2 and b = 2 5 (A) (2,3), (½, ²), (2,0), (0,0), (0,3) (D) (2,3), ( ½, ³), (2,0), (0,0), (0,4) (G) (3,2), (½, ²), (2,0), (0,0), (0,4) where (B) (3,2), (½, ½), (2,0), (0, 0), (0,3) (E) (2,3), (½, ½), (2,0), (0, 0), (0, 3) (H) (3,2), (½, ½), (2,0), (0, 0), (0, 4) (C) (2,3), (2, 2), (2,0), (0,0), (0, 4) (F) (3,2), (3, 1), (2,0), (0,0), (0, 3)

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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)