Let F: IR3 →IR3 is a diffeomorphism and M is a surface in IR3 , prove that the image F(M) is also a surface in IR3.

Q: Suppose the motion of a continuous body is given by x(X, t) = (X1 +tX2)en +(X2-tX1)ez + X3e3…

A: The motion described is by linear combinations. We can write the linear combination in the form of a…

Q: 1. Use the Divergence Theorem to evaluate F.dS where F (3z – za) i+(- 1) j+ (4y +) k and S is the…

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Q: Let C = C₁ U C₂ U C3, where C₁ is the line segment from (-1, −1) to (0, 0), C₂ is the line segment…

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Q: 3. Evaluate [[[ 20 av G if G is a solid in the first octant bounded by the plane y + z = 2 and the…

A: Solution :

Q: 3. Find points at which the mapping defined by f(z) = nz+z^n (n E N) is not conformal.

A: Given :  f(z) = nz+zn,  where  n∈ℕ. To find : The points at which f is not conformal.

Q: Find the image of the semi-infinite strip x ≥ 0, 0 ≤ y ≤ π under the transformation w = exp z, and…

A: w lies in the portion of the closed upper half-plane external to the open unit disk.

Q: Find Laplace ice transforms of the ictions: suonpun f(t) = t – 2e3t

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Q: 20. Find curl(F), where F(x, y, z) = xyz¹i+x²zªj + 4x²yz³k. a) (-4xyz³)j + (xz¹)k b) x²yj+xz¹k c)…

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Q: 1 : : Evaluate F dS whereF = (r",y",) and S is the solid bounded by the ry plane and the elliptic…

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Q: Use the Laplace transformation to solve :the problem ou du , x >0, t >0 u(0, t) = &t + 10 sinh 7t,…

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Q: 3. Use Stokes' Theorem to evaluate $ F dr, where F = (0, 0, 2), (1, 0, 2), (1, 1, 2), and (0, 1, 2),…

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Q: 3. Evaluate if G is a solid in the first octant bounded by the plane y+ z = 2 and the surface y = 1–…

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Q: Let f(x,y, z) = xy – yz and v [2y, 2z, 3x]. Find Curl(fv).

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Q: Let T:V U be a homomprphism then ker T={0}if T is one-one.

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Q: (c) Let z(x, y) = xr be a function defined on a disk D in the positive quadrant containing the point…

A: Follow the steps.

Q: Prove that C[x, y]/ is an integral domain

A: Gauss's lemma: A non-constant polynomial in ℤx is irreducible in ℤx if and only if it is both…

Q: 1)Prove that the curves o, = {|z|=2} y 0, = {|z- i|=4} are homotypic. 2)calculate O z*dz and o z*dz…

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Q: Let g, h: -R and /:R'+ R be defined by /(x, y) = (g(x, y), h(x, y). Assume curl /(x, y) = 7x – 7y,…

A: Let g,h:ℝ2→ℝ and f:ℝ2→ℝ2 and fx,y=g(x,y),h(x,y). Assume that ∇×f=7x-7y, ∇f=7y-8x and…

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: Let C = C₁ U C2 U C3, where C₁ is the line segment from (-1, −1) to (0, 0), C₂ is the line segment…

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Q: 4. (Section 17.8) Use the Divergence Theorem to calculate [[F F-dS, where F = y + ryj – zk and S is…

A: To use the divergence theorem to calculate ∫∫SF⋅dS.

Q: 9. If F = (x – z) î + (x³ + yz)ĵ + 3xy² k and S is the surface of the cone z = a – Va? +y?) above…

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Q: If F(x, y, z) = –2i – 2æj – yk and G(x, y, z) = –2xi – yj – 2zk, find curl(F × G).

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Q: 8) Under the bilinear transformation w=u+iv=iz, the image of the infinite strip 0<x< 1, is given by…

A: According to bartleby honor code and guidelines we have to do only first question

Q: (a) Find the line integral of f (x, y, z) = -x + y + z over the straight-line %3D segment from (1,…

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Q: 1. Let F(x, y) = (y, 2x2). Use the circulation form of Green's theorem to calculate the circulation…

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Q: bound the module of the contour integral (image) where C= { z E C : |z - 1| = 2}}

A: Introduction: Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement…

Q: (b) Show that the function defined by IN if (x, y) (0,0), √√√z²+y² f(x, y) = 0 if (x, y) = (0,0) is…

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Q: Find the residues of f(z) = z²-2z (z+1)²(z²+4) at all its poles in the finite plane.

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Q: Use the Laplace transformation to solve the problem: x > 0, t>0 u(0,t) = 3t + 8 sinh 4t, lim u(x, t)…

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Q: now that the mapping a: E? ->E, (x,y) Coy, x + 2 ) is a transformaton and a collineation by finding…

A: One-One: Let (x1,y1),(x2,y2)∈E2 be such that: α(x1,y1)=α(x2,y2) so,…

Q: Use the Laplace transformation to solve the problem: o'u o'u x>0, t>0 %3D u(0,t) 4t+10 sinh 1t, lim…

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Q: Let f(z) = 5z + 14, and let C1 be the line segment from z = 0 to z = 2, let C2 be the line segment…

A: In this question, we calculate the complex integral along the given contour joining vertices of the…

Q: 3. Evaluate AP G if G is a solid in the first octant bounded by the plane y + z = 2 and the surface…

A: The given problem is to evalute the triple integral over the given surface of first octant bounded…

Q: Please solve

A: Consider the given function.

Q: Let C = C₁ U C₂ U C3, where C₁ is the line segment from (-1, -1) to (0, 0), C₂ is the line segment…

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Q: 5. Verify Green's Theorem in the plane for F = (xy + y²)î + (x – y)f and the curve C is as shown in…

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Q: Let C = C₁ U C₂ U C3, where C₁ is the line segment from (-1, -1) to (0, 0), C₂ is the line segment…

A: The main objective is to show that F is conservative vector field. Find the potential function. Also…

Q: Show that the mapping w = (1 – j)z, where w = u + jv and z = x + jy, maps the region y > 1 in the z…

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Q: et X=R^(2) and C={(x y)in R^(2):x>=0 y=0}. Show that C is not a generating cone.

A: Given: Let   and .  To show that C is not a generating cone.

Q: Let the rectangular region R in z-plane which is bounded by the lines x= 0, y = 0,x= 2, y =1.…

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Q: Suppose C is a positively oriented, piecewise-smooth, simple closed curve. Verify that the result of…

A: Given    ∫Cxy2dx-x2ydy Where C is arc of parabola y=x2 from -1,1 to 1,1 and the line segment 1,1 to…

Q: 17. If f(x,y,z) and g(x, y, z) are continuously differentiable scalar functions defined over the…

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Q: How do I find the divergence and the curl of this?

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Q: Use Stokes's Theorem to evaluate F dr, where F(x, y, z) = x²i+z²j-xyzk and S is z = /169-x² -y² .…

A: As per our guideline we can't use any computer algebra. I calculate both line integral and surface…

Q: 8. Let F = 2 (in 2D). Show that F is divergence-free on its domain, that is, V.F= 0 at all points…

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Q: Let C = C₁ U C2 U C3, where C₁ is the line segment from (-1, -1) to (0, 0), C₂ is the line segment…

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Q: (a) Let D be the region in ryz-space defined by the inequalities 1 <r < 2,0 < ry < 2, 0 < z< 1.…

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Q: Let C = C₁ U C2 U C3, where C₁ is the line segment from (-1, −1) to (0, 0), C₂ is the line segment…

A: a. Suppose,  F=∇fF=∂f∂x,∂f∂y So, ∂f∂x=12x+2+tan-1y∂f∂y=x1+y2+3y2 Now, ∂f∂x=12x+2+tan-1y Integrate…