Show that fc F dr. is not independent of path. [Hint: Compute and , where  c1 and c2  are the upper and lower halves of the circle x 2 + y 2 = 1 from (1,0) to (-1,0)] Does this contradict Theorem 6?

Q: Can one draw two non-intersecting closed curves that each do not bound disks on RP ^2? How about on…

A: The real projective plane (RP^2 ) is classically defined as the space of lines through the origin in…

Q: Evaluate I = fe(sin z + 9y) dz + (7z + y) dy for the nonclosed path ABCD in the figure. А - (0,0),…

A: Given integral is I=∫Csinx+9ydx+7x+ydy For non-closed path, integration will be…

Q: Let C be the curve y = 2.5 In (6.25 – x²), for – 1.2 < x < 1.6. A graph of y follows. 4.6 4.5 4.4…

A: Here, y=2.5 ln6.25-x2. So,         y'=2.5ln6.25-x2'           =16.25-x26.25-x2'    =2.5·16.25-x2-2x…

Q: 5. Let C be the circle x +y} =4_which is positively oriented. Let C, be the upper half-circle C1 =…

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Q: Let C, be the straight line path z = (1 + i)t, 0 Sts 1; Cz the quarter circle z =1- cost +isin t, 0…

A: Given that C1 be the straight line path z=(1+i)t, 0≤t≤1; C2 the quarter circle z=1-cos t +i sin t,…

Q: Evaluate I = Se(sin z + 4y) dz + (5x + y) dy for the nonclosed path ABCD in the figure. %3! A =…

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Q: 4 and Consider the region in the second and third quadrants between the circles x² + y² = x² + y² =…

A: Concept: cartesian coordinates can be converted into polar coordinates using trigonometric functions…

Q: 2.2 Find upper bound of z + 4 -dz (z – 3i)3 + 1 where C is the circle |z – 3i| = 4.

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Q: Suppose that C = {z:|2| = 1} is the unit circle, followed in the positive (anticlockwise) direction.…

A: Given: C=z:z=1is the unit circle  f(z)=∫cdz4+3z

Q: Find the are length of the polav Curve r=3° from o-0 to 0= 1

A: Solve the following

Q: If C is the line segment which begins at (-2, 1) and ends at (2, 4), evaluate the line integral: et…

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Q: Can one draw two non-intersecting closed curves that each do not bound disks on RP2? How about on a…

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Q: Verify Green's Theorem in the plane for F = -3y³i+ (3x3 + cos y)ĵ where C is the circle x² + y² =…

A: In this we have to verify green theorem.

Q: If lx + my = 1 touches the curve (ax)" + (by)" = 1, show that %3D )+) = 1.

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Q: Determine whether the work done along the path C is positive, negative, or zero. y O positive…

A: Consider that the sign of the work depend on the orientation of the path C. here the circulation can…

Q: By using Green's theorem evaluate (4.ry – y') dr + (x + 3ry) dy where C is a closed curve starting…

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Q: Show that F(z, y) = (ry, z°y?) is not path independent by considering the line segment from (0,0) to…

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Q: Use Green's theorem to calculate where C is the rectangle with vertices (0, 0), (2, 0), (2, 3), (0,…

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Q: Demonstrate that the circles (x – 2)? + y² = 2 and x? + (y – 2)2 = 2 share tangent at point (1, 1),…

A: Here we will apply the concept of finding the equation of tangent to the two circles at point (1,1).

Q: Calculate both sides of Green's Theorem when F = [-x²y, xy²] and y positively oriented circle x2 +…

A: According to the given information it is required to calculate both sides of the Green’s theorem.…

Q: Prove that the following curves are intersecting with a right angle: 5y-2x+ y -x²y =0 2y+5x+ x* –…

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Q: Evaluate the line integral Sc (2x² − 3y) dx + xy dy, where the curve/path C is the line 3 y = 2/2 x…

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Q: Evaluate O (6y+x)dx+(y+2x)dy where C is the circle (x-2)²+(y-3)² = 4 Without using Green's Theorem.

A: We will find out the required value.

Q: (b) Suppose C is a curve of length N, and ||F|| < M, where M is some positive number. Prove that F.…

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Q: Evaluate f 3z2dz over C, where C is the upper half circle |z – connects points O and 1+i. -글히 %3D플…

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Q: 17. Evaluate OF . • dr where C is the circle x +y² = 1, z = 0 and F=yi+zj +xk.

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Q: If z = x + iy, then the inequality |z + il < 3 is a: A. Circle with center at (0,1) and radius 3. B.…

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Q: Examples Fid io av ed of the vegim borodkl by ke vegin Cuve and line hy mlars of double ifegrafin,…

A: Please refer the attached image for complete solution.

Q: Evaluate I = Se(sinz + 2y) da + (8r + y) dy for the nonclosed path ABCD in the figure. %3D А 3 (0,…

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Q: Find the DE of the family of plane curves described and sketch some of its members.

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Q: The tangent to the circle x2 +y2 +10 x+5 y= 60 at the point (2 ,4) is Select one: O None O 13 y +…

A: Given that:x2+y2+10x+5y=60

Q: Compute ,f for f(2) +6 + 6i, positively oriented. sin(32) and path 2 that is the boundary of the…

A: Let's see the method of integration. We use residue method to solve this integral.

Q: 4. Let C be the the circle ² + y² = 4 and Calculate the work(W = [,F dr when C is traversed in the…

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Q: Let C be the line segment connecting 0 to ni. Then f. 1-e2z dz =

A: we have to find integration

Q: Let C be the curve connecting (0,0,

A: Given that C be the curve connecting  (0, 0, 0) to (1, 4, 1) to (3, 6, 2) to (2, 2, 1) to (0, 0, 0).…

Q: . By sketching the traces on the ry-, xz- and yz-plane

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Q: Show that the curve In (2X-1) , n> i has no stationary point

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Q: Express the circle specified by a = 2 cos(0) y = 1+2 sin(0) in Cartesian coordinates and find the…

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Q: Q5. Prove that the circle {(x, y), z² + y² = 1} is not homeomorphic to the interval (–1, 1). 1

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Q: Use Green's Theorem to evaluate -1 (y? tan a?) dx + (3.xy + 4x – In(y² + 4)) dy, where C is the…

A: The green's theorem states that "line integral over a closed curve C…

Q: Find the upper bound of |[∫c ez^2 + 1 / (z - 1)2 + 5] dz|, where C is the circle |z - 1| = 2.

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Q: 4. Prove that the following curves are intersecting with a right angle: 5y-2x+y²-x²y = 0 2y +…

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Q: Evaluate the line integral 3ydx + 6xdy where C is the straight line path from (1, 2) to (6, 5). Sc…

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Q: Use Green’s theorem to evaluate the line integral 2x2y dx + 6x3 dy, where C is the circle x2+y2 = 4,…

A: The answer is as follows:

Q: Use Green's Theorem to evaluate $. xydx + x²y³dy, where C is the triangle with vertices (0,0), (1,0)…

A: Consider the provided question, ∮cxydx+x2y3dy,     where c is the triangle with vertices 0,0, 1,0…

Q: Evaluate the line integral ∫Cx2yds along the segment of the line y=x from the origin up to the point…

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Q: Let C be the curve connecting (0,0) to (9,3) to (-3,3) to (0,0) using straight lines. Evaluate (sec…

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Q: 5. Show that the following orthogonal families of curves: are x² + y² : = ax, x² + y? = by

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Q: Let C be the circle |z| = 4 traversed once in the positive sense (counter %3D clockwise). Compute :…

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