16.11. Let y be the unit circle [2] = 1 traversed twice in the clockwise direction....
The circuit C, depicted below, is traversed clockwise starting and ending at -1. It consist of two parts: C = A + B where A c {z = z + iy : y = 1-2 and (a) Give parametrisations of A and B (b) Evaluate the line integrals L,-/ Re(z) dz, L,-, Re(z) dz. 1+2 (1+z+ z2)2 Calculus for complex line integrals to evaluate (c) Let f be given by f(z) - 2% Use the Fundamental Theorem of (d) Does...
(b) True or false: if I is the unit circle traversed once counterclockwise, then J. z dz - Adi You must justify your answer to receive full credit.
1 5. Let A = dz, (2 – 1)2(2 + 2i)3 where I is the circle [2] = 3 traversed once counterclockwise. The following is an outline of the proof that A = 0, justify each statement. Jo Tz – 1)*(x + 2133 (a) For R > 3 show that A = A(R) where A(R) Som 1 (z – 1)2(x + 2i)3 dz, and I'R is the circle (2|| = R traversed once counterclockwise. 21R (b) For R > 3...
Question 5 Evaluate where is each of the following contour. (a) is the path from (1, ) to (0-1) along the unit circle (centered at angin) in counter-clockwise direction. (b) C is the straight line from (1, 0) to (0-1). (c) C is the path along the square with vertices at (1.11) traversed in the clockwise direction (d) is the path along the circle of unit radius centered at (1.1) traversed in the counter-clockwise direction
Problem 1 (10). Let C denote the circle 2| = 2 in the positive direction. Evaluate the integrals. 4.50 (a) (2-3 2ds C (2) ee2 (b) peli-2)
Problem 1 (10). Let C denote the circle 2| = 2 in the positive direction. Evaluate the integrals. 4.50 (a) (2-3 2ds C (2) ee2 (b) peli-2)
2. (3 pts.) Let C denote the unit circle, oriented clockwise. Evaluate the line integral ydx dy in two different ways: first by parameterizing the curve and using the definition of line integral; then, use Green's theorem.
2. (3 pts.) Let C denote the unit circle, oriented clockwise. Evaluate the line integral ydx dy in two different ways: first by parameterizing the curve and using the definition of line integral; then, use Green's theorem.
F(x,y) =<2xy,x^2+y^2> the part of the unit circle in the
first quadrant oriented counter clockwise
37. F(x, y) = (2.xy, x2+y2), quadrant oriented counterclockwise the part of the unit circle in the first
37. F(x, y) = (2.xy, x2+y2), quadrant oriented counterclockwise the part of the unit circle in the first
(b) Let C be the closed curve formed by intersecting the cylinder x2 +y= 1 with the plane x z= 2. Let the tangent to the curve from above. point in the anti-clockwise direction when viewed Calculate the line integral (e (e sin y+ 4) dy+(e(cos z+ sin z)+ay) dz. cos x2yz) dx +
(b) Let C be the closed curve formed by intersecting the cylinder x2 +y= 1 with the plane x z= 2. Let the tangent to the...
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8 pts.) 7. Find all solutions of in log 2 [12 pts. 8. Compute z dz where T is the upper half unit circle centered at the origin parameterized in the counter- clockwise direction.
8 pts.) 7. Find all solutions of in log 2 [12 pts. 8. Compute z dz where T is the upper half unit circle centered at the origin parameterized in the counter- clockwise direction.
1. Consider the unit circle: (x,y) : x2 y2 = 1. T. Let f R2 ->R be defined by f(x,y) = x2-y, and let F : R2 -> R be defined by F(x, y) Compute the integral of f and F around the unit circle. For the integral of F, proceed in the standard (anticlockwise) direction