Let C be a circle with |z - i| = 3 travelled once in counterclockwise direction. Evaluate: 5z3 + 3z2 – 3z – 5 -dz (z – 2i)3

Q: COSTZ (iii) dz, where C is the Jc (z²-1)2 circle |z – 2| = 2 counterclockwise.

A: Let fz=cosπzz2-12 and C is the circle z-2=2 fz can be written as fz=cosπzz2-12=cosπzz-12z+12…

Q: Let F = (5e**+1 + y)ï + (2x – sin(3y – 2))j and C be the boundaries of the circle x? + (y – 2)² = 4…

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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: 2. Find the contour integral f,zdz for (a) y is the triangle ABC oriented counterclockwise, where A…

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Q: 10. Let C denote the positively oriented circle z| = 1. Show that - Ti 2sin(z) dz cos(z) dz 4 z(z²…

A: See the attachment.

Q: 4. Consider the curve in the plane defined by r3+y3 = 6xy. %3D (a) Find for any point (r, y) on the…

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Q: Let f (z) = SLet C'be the circle |z| = 2in the counter-clockwise direction. What is | f(2) dz 2ri C

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Q: Verify greens theorem form 1 for F=x^3i+yj and c is the circle x^2+y^2=1, oriented anti clockwise

A: Find ∫CFdr, the line integral of F, where C is the circle x2+y2=1, oriented anti clock wise…

Q: Let C be the curve of intersection of the parabolic cylinder x² = 2y, and the surface 3z = xy. Find…

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Q: 2. Let C₁ denote the positively oriented boundary of the square whose sides lie along the lines x =…

A: Given: Let us consider the given data, The lines x=±1y=±1 Circle,z=4 To show ∫C1fzdz=∫C2fzdz a) when…

Q: Sketch the plane curve r(t) = t2i − 4tj and find its length over the given interval [0, 5] .

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Q: use green's theorem to evaluate ∫c 5y^3dx-5x^3dy where C is the circle x^2+y^2=4 counterclockwise

A: Given: ∫c 5y^3dx-5x^3dy where C is the circle x^2+y^2=4

Q: 6. Evaluate [(x + y²)dx – xdy where C is the circle "x²+y² : = 1" with a counter- clockwise…

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Q: =2) Calculate & ydx + xz² dy - zy³ dz, where c is the circle x² + y² = 4, Z=3 (conterclockwise)…

A: We will use stokes theorem to find integral

Q: Prove that the circle {(x, y), x² + y² = 1} is hot homeomorphic to the interval [-1, 1].

A: We will prove this result by contradiction: Suppose there exists a homeomorphism between -1,1 and S…

Q: 3. a) Compute: f (Rez-3z) dz where I is a directed line segment from z = 0 to z = i. b) Let C be the…

A: To Evaluate:  (a) ∫Γ(Rez-3z)dz , where Γ: z=0 to i. (b) ∮Ce-zz2(z-5)dz, where C: |z|=4 oriented…

Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

A: Greens theorem states that "for closed curve C enclosed a region R then the line integral along the…

Q: Let C be the circle |2- i 2 oriented counterclockwise. Evaluate the following integral 1. z(2² + 4)

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Q: Describe the projections of the circle r(t) =〈sin t, 0, 4 + cost〉onto the coordinate planes.

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Q: A D E y 50 W V 20 70 В ZDCE= 90 AD = AE = AC = AB by the Hypotenuse Midpoint Theorem + Thus DECB is…

A: The given figure is- Given that AD=AE and AB=AC

Q: Let F = (tan-(3x) – y)ỉ + (-2x + sec¬1y)j and C be the boundaries of the circle x? + (y – 4)² = 9 in…

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Q: zt e dz if t>0 and C is the circle z]=2. Evaluate 2ni C 2πίz+1

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Q: Use the Green's theorem to calulate So (3x²y – xy? – 3.x²y²) dx + (x³ – 2.x³y+ x²y) dy, where C is…

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Q: Evaluate the line integral ſe ry²ds, where C is three quarters of the circle a² +y? = 1, oriented…

A: C is the three quarters of the circle x2+y2=1, oriented counterclockwise. The parametric equation of…

Q: Find the extreme values of the function ƒ(x, y, z) = xy + z2 on the circle in which the plane y - x…

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Q: Let C be the intersection curve of the parabolic cylinder √2x2 = 2y and the surface 3z =√2xy. Find…

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Q: Find a parametrization c(θ) of the unit circle such that c(0) = (−1, 0).

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Q: dA, where K is the disk r + (y 2 +?

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Q: (b) Find eiz dz- (z + i)3 Izl=1.5 (the circle is oriented counterclockwise).

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Q: Let f (z) -Let C'be the circle z 10in the counter-clockwise direction. What is 1 -Lf(2) dz? 2ri

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Q: (b)for the space curve,r(t) = costi + sintj + tk, 0st< 4n

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Q: Jse Green's theorem to evaluate $ (2°e* + 4y) dz + (-5x+ y* sin(y)) dy, where C is the…

A: The given path C is circle x2+y2=16. Therefore: D= (x, y) : x2+y2≤16

Q: i) Use the residue theorem to evaluate 2z+6 dz, where c is the circle |z – i| = 2. z²+4

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Q: Find the curvature k of the circle C : (x – 2)² + (y+ 3)² 16 at the point (2, 1).

A: Use the formula of curvature to solve this problem.

Q: Let F (tan-(3x) + y)i + (2x + sec-ly)j and C be the boundaries of the %3D circle x? + (y- 4)2 = 9 in…

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Q: (a) Find eiz dz. Jt-0,5 (z+ i)3 (the circle is oriented counterclockwise).

A: Here we use the Cauchy residue theorem to find the integral. But here the given function is analytic…

Q: Let C' be the circle of equation a² + (y - 2)2 in counterclockwise direction. a) Find the…

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Q: EVALUATE § 7. d² USING STOKE'S THEOREM 3xz^ + ex² ₁ + 2xy^ с F(x, y, z) = 3XZ1 te C IS THE CIRCLE…

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Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

A: Greens theorem

Q: Find the curvature K of the circle C: (x - 2) + (y+3)² = 16 %3D at the point (2, 1).

A: Given: Equation of circle C: (x-2)2+(y+3)2=16 ⇒ (x-2)2+(y-(-3))2=42 We know that circle equation is…

Q: Let F = -2yi + 2xj. Use the flux form of Green's Theorem to compute F·n ds where C is the positively…

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Q: Let C be the positively oriented circle x² + y² = 1. Use Green's Theorem to evaluate the line…

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Q: Q5. Prove that the circle {(r, y), x² + y? = 1} is not homeomorphic to the interval [-1, 1].

A: Here we can prove this question using topological property connectedness.

Q: 5. Let C denote the positively oriented circle z- 1= 3. Evaluate the integral: 3 z +2z dz 3. (z- 2)

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Q: Evaluated the counter integral 9,+z² - z-1 dz ,where C 1. The circle z| = 0.5 2. The circle |z+1-i=…

A: Given contour integral is  ∮Cezz3+z2-z-1dz We can write it as ∮Cez(z+1)2(z-1)dz Poles of…

Q: Let C be the curve connecting (1,1) to (-1,-1) clockwise around the circle x2 + y = 2 1 -dy + y? + 1…

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Q: Let C be the curve of intersection of the parabolic cylinder x? = 2y, and the surface 3z = xy. Find…

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Q: Sketch the plane curve r(t) = t2i + 2tk, and find its length over the given interval [0, 3]

A: We have, r(t)=t2i+2tk  we have to determine the length of the curve over the interval [0,3]

Q: (b) cos(z) -dz, z2 + 2iz – 1 where y is the circle {|z + i| = 1} with counter clockwise orientation.

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