IUT Problem ? (2 points). Give an explicit example of a homeomorphism. f: [0,1] → [0, 2].
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A: Please find attachment. Degree of characteristic equation is equal to number of roots.
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A: Please find attachment
Q: part D E F
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Q: Let k and a be arbitrary integers. Prove that a is even if and only if ka is even.
A: SEE BELOW FOR TGE COMPLETE SOLUTION
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Q: Can you please in words explain how you got the answer that you did?
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Q: The system x' + y = t; 4x + y' = 0; where: x(0) = 1; y(0) = 2 has a -2t (x = a + be ly=t+de + ce²t…
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- Let F (8xy, 3y, 8z). = The curl of F = (000). Is there a function f such that F = V f? ☐ (y/ (y/n)Let θ : R^4 → R defined by θ(u) = sup{u1|, |u4|}. Determine if θ is a norm, a seminorm or none of the two for R^4Show that the following are not llinear transformations, explain why. T(x,y) = (x^3, xy, y) T(x,y) = (x-2, y+2)
- ExL f.9) = dx (f.9)3D (8x-5) f(x).g(x) of oll continous Let v be the vector real valued function on the intoval [o,] space nterva c(co,2]) Examine th at the function example I is an Note: Show steps using f înnor prodct Space n defined n înner product or not 2 definiton u Sin a ce înnerFind L{f (t)}using the First Shifting Theorem, Note: a, b, e, w, k and n are constants 1. Il aydc30329 2. L{e* (acost+6 sin t)} 3.Compute the divergence and curl of F= -(z³-3x)j+4y²k+x²yi A. correct divergence not among the choices OB. divergence = 2xy OC. divergence = (8y+3z²)i+ (3-x²)k D. curl = (8y+3z²)i+ (3+x²)k E. correct curl not among the choices F. curl = 2xy
- (4) Let ƒ₂ = 0ƒ/əz and ƒz = 0ƒ/əz. Recall f = u + iv, the notation f is used for the complex conjugate function u - - iv. (d) The Jacobian of f, viewed as a map U → R², is defined by Jf = det (Df). Show that Jf = |fz|² - |fz|². In particular, if ƒ € O(U), then Jƒ = |ƒ'|².(a) Let u be a function of R2 such that ди ди an(x, y) = (x, y); for all(x, y) ∈ R2 ду then prove that au (x,y") ду u(x, y) – u(y, x) = (x-y) (x*, y*) + (y-x) ди Әх for some point (x*, y*) ∈ R2 (b) Evaluate the general solution of partial differential equation (x−y)ux+(y−x−u)uy =u