Draw the set of all points (z, y) E R2 satisfying the following equations: 5. xy...
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
1. Sketch the following set of points in the z-y plane: {(x, y) € R2 :(y - x²)(y + |21) >0}
3. Describe geometrically the set of points z E C satisfying (a) 10 points ziz -i
Let f : R2 + R be defined by f(x,y) = |xy|e-(z?+y?). Evaluate SR2 f, if it exists
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that: 1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
1.29. Let G be the set of points zeC satisfying either z is real and -2 <z<-1, or lz< 1, or z 1 or z = 2. C (a) Sketch the set G, being careful to indicate exactly the points that are in G. (b) Determine the interior points of G ELEMENTARY TOPOLOGY OF THE PLANE 2I (c) Determine the boundary points of G. (d) Determine the isolated points of G. 120 TL 1.29. Let G be the set of...
2. You are given the following multivariate PDF 0 else where S = {(z, y, z) |エ2 + y2 + 22-1). (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inseribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the probabilities P[(X, Y, Z)...
For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded invariant set of the dynamical system (in fact, it's also closed) (3) Find all the equilibria of this system. Show that H-() is made up of one equilibium point and two homoclinic orbits attached to it. (4) Sketch the invariant...
Let A C R and fA: R2-given by 1 if (x, y) E A 0 if (r, y) A Ar, y): a)Prove that fAis continuosin int(A)Uert(A) and f is dicontinuos in cl(A) b)Draw fA a) A = B2 (0) . b) A = {(x,y) | xy = 0} . c) A = {(z, y) | y E Q)