Are |z], Img(z) analytic? Give reason. sin Choose the suitable theorem to find the result of $ dz over ntegration path C. where C any contours in z-plane and z = 0, ±1, ±j

Q: Sketch the space curve and find its length over the given interval r(t) = ⟨8 cos t, 8 sin t, t⟩,…

A: Given Finding the derivative with respect to t:

Q: Let F(x, y) = x2 + x² + y2J- (a) Show that F. dr +| F. dr if C and C2 are the semicircular paths…

A: The function given to us is  F(x,y)=yx2+y2i-xx2+y2j

Q: If curl F = (2² + z²)j+5k, find ſař · dř, where C is a circle of radius 3, centered at the origin,…

A: Given curlF=(x2+z2)j+5kTo find ∫CF.drwhere C is cricle of radius 3.

Q: Find a piecewise smooth parametrization of the path C. (ti + tj 0 sts 1 r(t) = 1st< 2 y = VI (1, 1)…

A: C:y=√x

Q: Find a mat rix P such that p-l 1 0 1 0 %3D 3 2 -1 0 P%3D OA. 3. 13 P = В. 0 1 1 0 P = C. 3 1 1 0 [1…

A: In the question we have to find P matrix.

Q: Evaluate the contour integral. (complex analysis) a) 1 dz where C is the semicircle |z – 1| = 1, Im…

A:

Q: The line integral P F.d7 can be evaluated using Stoke's Theorem where F = (2x, -3, 1) and C is an…

A:

Q: Evaluate the contour integral using the parametric representation for C, where 22 – 1 f(2) and the…

A:

Q: Use Stoke's Theorem to evaluate $. F · dR, where F (x, y, z) = (z²,xy, xz) and C is the triangle in…

A:

Q: Find T, N, and k for the space curves r(t) = (cosh t)i - (sinh t)j + tk

A: Let the given space curve is r(t)=(cosh t)i-(sinh t)j+tk   To determine: The values of the unit…

Q: 1. Let u be a smooth harmonic function inside the unit ball B(0, 1) in R? so that |u| < 1. Prove…

A: To prove this we will use the following tools : Solution of Laplace's equation in polar coordinates…

Q: Use Stokes's Theorem to evaluate JeF - dr- . In each case, C is oriented counterclockwise as viewed…

A:

Q: Let u(x,y) = e" (xcos y-y siny) (a) Show that u(x,y) is a harmonic function. (b) If v(x,y) is a…

A:

Q: Prove that the Lagrange's fomula can be put in the form ø (x) f (x,) (x- x,) ø' (x,) P„(x) = £ %3D |…

A:

Q: 1. Evaluate z dz, where the contour C goes from z = 0 to z = 1 to z = 1+i.

A:

Q: Let z be a complex mumber , f() b any funcion, and c be contour any then to f B(Z)dZ?

A: Let z be a complex number, f(z) be any function and c be any contour.     A contour is a type of…

Q: The Real Part of the cor u(x,y (i) Prove that u(;

A:

Q: 4. Evaluate fo F• dr by the Stoke's theorem, where F = -y? î+x j+ z² k, and C' is the curve of…

A: Given :

Q: Prove that the Lagrange's fomula can be put in the form ø (x) f (x,) (x- x,) ø' (x,) n P„(x) = Ë r =…

A:

Q: Prove that | f(2)dz → ViRes(f, zo) 8 → 0. by letting Is = {z = 20 + de®; 0<0<¥} %3D where f is…

A: Given: Γδ=z=z0+δeiθ; 0≤θ≤Ψ where f is analytic in a neighbourhood Bz0, r except for a simple pole at…

Q: Suppose that (z) is entire and that the harmonic function u(x, y) = Re[fz)] has an upper bound uo;…

A: As requested following is the answer for question 12.

Q: Find the first and second fundamental form of x(u,v)=(e(u-v)/2 cos(u+v)/2, e(u-v)/2…

A:

Q: = Let Co be the curve formed by the intersection of the surfaces x₁ = 2x² and X3 3x1x2, and let C be…

A:

Q: Sketch the space curve r(t) = a cos ti + a sin tj + btk and find its length over the given interval…

A: Given : Space curve equation rt = acosti^ + asintj^ + btk^  ,   and  t ∈ 0,2π  ,a and b are contants…

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…

A:

Q: Find T, N, and k for the space curve r(t) = - ti - (a cosh (t/a))j, a > 0.

A: Unit Tangent. Unit Normal. Curvature.

Q: Consider the function f (z) = e8? (a) Use the Cauchy-Riemann equations to show that f (z)is not…

A: Analytic function,. C-R equations

Q: The DE: y" + P(x)y' + Q(x)y = 0 in which both terms P(X) and Q(x) are analytic at X = X0, then the…

A: From  Statement

Q: 3. Sketch the space curve, r(t) = 8 cos ti +8 sin t j+tk , and find its length over the interval []…

A: Given: r(t)=8cost+8sint+tk    [0,pi/2]

Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

A:

Q: Given F=. By using Gauss's theorem, find f F ds for a paraboloid x = 4-y² - 2² with x ≥ 0 as shown…

A: We are given  F=-3x+1+z3,y+1+x3,6z+xy So, the divergence of F is div…

Q: Use the Cauchy-Riemann conditions to find out whether the following functions are analytic. b.…

A:

Q: Evaluate f Im(z – i)dz where C is the contour from z =1+ 2i to z z = i to z = -i.

A:

Q: Find T, N, and x for the space curve r(t) = 7ti + (7a cosh (t/a))j, a > 0. T(t) = (Di+ (Di (Type…

A: According to the given information, it is required to find T, N and k for the space curve r(t).

Q: (a) Let f(z) = %3D (z + 4- i) (z4 + 5z³) Using CRT, evaluate the integral Of(2)dz, where the contour…

A: In the given question we have to evaluate the integral                               where the…

Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

A:

Q: Sketch the space curve r(t) = ⟨4t, −cos t, sin t⟩ and find its length over the given interval [0,…

A: Consider the given space curve, rt=4t,-cost,sint use the graphing utility to graph the curve,

Q: use Stoles's Theorem to evaluate F ds where Ě= and C is the orde with rading 5 centered C2,0,0…

A: According to Stokes' law:                                                     ∮JF·dr=∬S(Curl F·N)dS…

Q: The Cauchy-Riemann equations for f(z) is dv ду ду ду ди ду ди ди dx ду ди dv dx ду %3D %3D

A:

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

A:

Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

A:

Q: B) Use Green's Theorem to evaluate /F-dr, where C is a triangle from (0,0) to (1,0) to (0,3) to…

A: We will use Greens theorem here

Q: Use Green's Theorem to evaluate the line integral  ∫c (y-x) dx + (2x-y) dy for the given path. C:…

A: Given data-    The line integral -                          ∫c (y-x) dx + (2x-y) dy .    We have to…

Q: Use Stoke’s theorem to evaluate ꬹϲ F.dr, if F(x,y,z)=(-xy,)I –(xz)j+(2yz)k and C is the closed…

A: The objective is to use the Stokes' theorem to evaluate the integral ∫CF·dR where…

Q: Q4: State the Fundamental Theorem of Independent of Path and Morera's Theorem. Why can't apply these…

A:

Q: b. What is fo, f (z) dz?

A: f(z) is holomorphic except at z0   where f(z) has pole at z0 we can write  f(z)=g(z)z-z0  where g(z)…

Q: Using Green’s formula, evaluate the line integral ∮C(x−y)dx+ (x+y)dy, where C is the circle…

A: