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- Evaluate [[cos (x-y)sin(x+y)dA, where R is the triangle with vertices (0,0), R (-7,7), and (7,7) using the change of variables u = x-y and v=x+y.(4) Decide which of the following maps f: R² → R² answers. (iii.) f(x, y) = (x+y, 0); are isometries. Justify your#24 ca] Show that f: IR→IR, x-(1-x)+cos(1+x³) is continous. b. Show that the map f: IR→IR xx²x хохе (2 - sinx) is continous.
- (4) Decide which of the following maps f: R2 R2 are answers. isometries. Justify your (iv.) f(x, y) = (2x+3y,+2);2. can you please do this step by step so I can understand it and follow along(4) Let f₂ = df/az and ƒz = df/az. Recall ƒ = u + iv, the notation f is used for the complex conjugate function u — iv. (b) Verify the chain rule (fog) z (fz °9)9z+(fz° 9)āz (fog)z = (f₂ °9)9z+ (fz°9)9z (fzᵒg)9z+ =
- Find F. dr for given F and C. (a) (2,4). F = -y sin(x)i + cos(x)] and C is the parabola y = x² between (0,0) and (b) F = 2yi – (sin y)] counterclockwise around the unit circle C starting at the point (1,0).Exercise 2. Find all fixed points and their stability for the following maps (use numerical or graphical arguments if necessary): • In+1 = %3D 1+*n • In+1 = 1+ sin xn1. Evaluate ("+") and (4,4), using change the of variables u = "ty, and v= = zzy JS₁ R sin COS (2) dA, where R is the triangle with vertices (0,0), (,0),
- (4) Decide which of the following maps ƒ R2 R² answers. (i.) f(x,y)=(y, x); -> are isometries. Justify your(8) If f(z) is analytic function expressed in polar form, show that: of (sin+icos) of Ər r ᎧᎾ f'(z)=(cos-isin). =Suppose that the temperature T at point (x, y, z) is given by T(x, y, z) = sin(xyz) and that an insect currently at the point (5, 0, 8) moves slightly in the direction parallel to the positive y-axis. As a result, the bug will feel the temperature... decrease stay the same increase