(c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be...
(e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto B. (0,1) (1;1) (-1,1) (0,1) v=1 1/2 y-axis y=x2 v-axis v -u b ets t) ide ods.a notteog (0,0) X-axis u-axis (0,0) (e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto...
5. Let f R2 ->R2 be the function given by f(x, y) (х + у, х — у). (i) Prove that f is linear as a function from R2 to R2. (ii) Calculatee the matrix of f. (iii) Prove that f is a one-to-one function whose range is R2. Deduce that f has an inverse function and calculate it. (iv) If C is the square in R2 given by C = [0,1] x [0, 1], find the set f(C), illustrating...
Question 19: Linear Transformations Let S = {(u, v): 0 <u<1,0 <v<1} be the unit square and let RCR be the parallelogram with vertices (0,0), (2, 2), (3,-1), (5,1). a. Find a linear transformation T:R2 + R2 such that T(S) = R and T(1,0) = (2, 2). What is T(0, 1)? T(0,1): 2= y= b. Use the change of variables theorem to fill in the appropriate information: 1(4,)dA= S. ° Sºf(T(u, v)|Jac(T)| dudv JA JO A= c. If f(x, y)...
3. Let S be the triangle with vertices at (0,0), (1,0) and (0,1). Let f (x, y) = e***. Use the change of variables u = x – y, v = x +y to find . f(a,y) dA.
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R defined by f(r,y)-+ (a) Show by explicit computation that the directional derivative exists at (x, y)- (0,0) for all oi rections u є R2 with 1 11-1, but that its value %(0.0) (Vf(0,0).u), fr at least one sucli u. (b) Show that the partial derivatives of f are not continuous at (0,0) if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R...
l maps is a quotient map. 4, Let ( X,T ) be a topological space, let Y be a nonempty set, let f be a function that maps X onto Y, let U be the quotient topology on induced by f, and let (Z, V) be a topological space. Prove that a function g:Y Z is continuous if and only if go f XZ is continuous. l maps is a quotient map. 4, Let ( X,T ) be a topological...
(b) Let F: R2 + Rº be a vector field on R2 defined as F(x, y) = (3y, 22 – y). Suppose further that ^ C R2 is a curve in R2 consisting of the parabola y = 22 - 1 for 1 € (-1,0) and the straight line y = 1 – 1 for 1 € [0,1]. (i) Sketch the curvey in R2 [2] (ii) By considering the curve y piecewise, compute the vector field integral: [5] F(x). F(x)...
Problem 3: Let f: X -> R, XC R2, be given by f(x, y)n(x 2y 1), V(r,y) e X Find the maximal domain X and write the second-order Taylor polynomial for f around the point (2,1) E X. (6 points) Problem 3: Let f: X -> R, XC R2, be given by f(x, y)n(x 2y 1), V(r,y) e X Find the maximal domain X and write the second-order Taylor polynomial for f around the point (2,1) E X. (6 points)
3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin, and let S : R2 + R2 be the shear along the y-axis given by S(x,y) = (x,x+y). (You may assume that these are linear transformations.) (a) Write down, or compute, the standard matrix representations of T and S. (b) Use (a) to find the standard matrix representations of (i) SoT (T followed by S), and (ii) ToS (S followed by T). (c) Let...