| № |
Condition |
free/or 0.5$ |
| m624 | Let f: R2 -> R be defined as in Problem 1-26. Show that Dxf (0, 0) exists for all x, although f is not even continuous at (0,0). |
buy |
| m625 | Let f : R2 ->R be defined by
f(x, y) = { x|y| / √x2 + y2 (x, y) ≠ 0, 0 (x, y) =0. |
buy |
| m626 | Let f : R2 ->R be defined by f (x, y) = |xy|. Show that f is not differentiable at 0 |
buy |
| m627 | Let f: R->R 2. Prove that f is differentiable at a € R if and only if f 1 and f 2 are, and in this case
f 1(a) = ((f 1)1 (a) (f 2)1 (a)). |
buy |
| m628 | Let f: Rn ->R be a function such that | f (x) | ≤ |x|2 . Show that f is differentiable at 0 |
buy |
| m629 | Let f: Rn ->R. For x€ Rn,
a. Show that Deif (a) = Dif (a)..
b. Show that Dtxf (a) = Dxf(a)..
c. If f is differentiable at , show that Dxf(a) = Df(a)(x) (a) and therefore Dx + yf(a) = Dxf (a) + Dyf (a).. |
buy |
| m630 | Let g: A ->Rp be as in Theorem 5-1. If f: Rn -> R is differentiable and the maximum (or minimum) of f on g-1 (0) occurs at , show that there are , such that |
buy |
| m631 | Let g1, g2: R2 -> R be continuously differentiable and suppose D1 g2= D2 R1.. As in Problem 2-21, let |
buy |
| m632 | Let g1, g2: R2-> R be continuous. Define f: R2->Rby f(x,y) =
(a) Show that D2f (x,y) = g2(x,y)
(b) How should f be defined so that D1f(x,y) =g1(x,y)?
(c) Find a function f: R2->R such that D1f (x,y)=x and D1f (x,y)=y |
buy |
| m633 | Let Kn = {xЄRn: x1 = 0 and x2. . . x n−1 > 0}. If MCKn is a k-dimensional manifold and N is obtained by revolving M around the axis x1 = . = xn-1=0, show that N is a (k + 1) -dimensional manifold. Example: the tours (Figure 5-4). |
buy |
| m634 | Let M be an (n – 1) -dimensional manifold in Rn. Let M (Є) be the set of end-points of normal vectors (in both directions) of length Є and suppose Є is small enough so that M(Є) is also an (n- 1)-dimensional manifold. Show that M(Є) is orientable (even if M is not). What is M(Є ) if M is the M"{o}bius strip? |
buy |
| m638 | Let U be the open set of Problem 3-11. Show that if f = X except on a set of measure 0, then f is not integrable on [0, 1] |
buy |
| m737 | Prove that a k-dimensional (vector) subspace of Rn is a k-dimensional manifold. |
buy |
| m738 | Prove that if f: Rn -> Rm is differentiable at a € Rn, then it is continuous at a. |
buy |
| m739 | Prove a partial converse to Theorem 5-1: If MCRn is a k-dimensional manifold and xЄM, then there is an open set A C Rn containing and a differentiable function g: A ->Rn-k such that A∩M = g-1 (0) and g1 (y) has rank n – k when g(y) = 0. |
buy |
| m740 | Prove that R = [a1, b1] x..x [an, bn] is not of content if ai < bi for i =1. n. |
buy |
| m822 | Regard an n x n matrix as a point in the -fold product Rn x . x Rn by considering each row as a member of Rn..
a. Prove that det : Rn x . x Rn -> Rn is differentiable and
b. If aij : R ->R are differentiable and f(t) = det (aij(t)), , show that |
buy |
| m858 | Show by induction on n that R = [a1, b1] x..x [an, bn] is not a set of measure 0 (or content 0) if ai < bi for each i. |
buy |
| m860 | Show that if C has content 0, then C C A for some closed rectangle A and C is Jordan-measurable and ∫ AXC = 0. |
buy |
| m861 | Show that if f, g: A -> R are integrable, so is f ∙ g. |
buy |