- Prove that if f: Rn → Rm is differentiable at a € Rn, then it is continuous at a.
- A function f: R2 → R is said to be independent of the second variable if for each x € R we have f (x, y1) = f (x, y2) for all y1, y2. €R Show that f is independent of the second variable if
- Define when a function f: Rn → R is independent of the first variable and find f1 (a, b) for such f. Which functions are independent of the first variable and also of the second variable?
- Let be a continuous real-valued function on the unit circle {x € R2 : |x| =1} such that f (0, 1) = g(1, 0) = 0 and g(- x )= - g(x). Define f: R2→R by F (x) = {|x| . g (x/|x| x
- Let f : R2 →R be defined by f(x, y) = { x|y| / √x2 + y2 (x, y) ≠ 0, 0 (x, y) =0.
- Let f : R2 →R be defined by f (x, y) = |xy|. Show that f is not differentiable at 0
- Let f: Rn →R be a function such that | f (x) | ≤ |x|2 . Show that f is differentiable at 0
- 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)).
- Two functions f , g : R → R are equal up to nth order at if lim h → o f(a + h) – g(a + h)/hn =0 (a). Show that f is differentiable at if and only if there is a function g of
- Use the theorems of this section to find f1 for the following: a. f(x, y, z) = xy b. f(x, y) = sin (xsin (y)). c. f(x, y, z) = sin (xsin (ysin (z)) d. f(x, y, z) = xy2 e. f(x, y, z)
- Find f1 for the following (where g: R → R is continuous): (a) f (x, y) = g (b) f(x, y) = g (c) f(x,y,z)=
- A function f: Rn x Rm→ Rp is bilinear if for x,x1, x2 € R n, y,y1, y2 € Rm and a € R\ We have, f(ax, y) = af (x, y) = f(x, ay) f(x1 + x2, y) = f(x1, y) + f(x2, y) f(x, y1 +y2) = f(x,
- Define IP: Rn x Rn →R by IP (x, y) = . (a) Find D(IP) (a,b) and (IP)’ (a,b). (b) If f,g: R → Rn are differentiable, and h: R → R is defined by h(t) = , show that hI (a) =
- Le Ei, i = 1,., k be Euclidean spaces of various dimensions. A function f: E1 X. X Ek→Rp is called multi linear if for each choice of xj € Ej, j ≠ I the function f: Ei→Rp defined
- 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
- Suppose f: Rn →Rn is differentiable and has a differentiable inverse F -1: Rn → Rn. Show that (f-1) I (a) = (fi (f-1(a)))-1.
- Find the partial derivatives of the following functions: a. f(x,y,z)=xy b. f(x,y,z)=z c. f(x,y)=sin (xsin (y)) d. f(x,y,z)= sin (x sin (y
- Find the partial derivatives of the following functions (where g: R →R is continuous): (a) f(x,y ) = fx+ y g (b) f(x,y ) =fx g (c) f(x,y ) =f xy g (d) f(x,y ) =f(fyg)g
- f (x,y) = xxxxx = (log (x))(aretan (aretan(sin(eos(xy)) – log (x + y ))))),
- Define f: R → R by f (x) = { e-x-2 x ≠ 0,. 0 x -0,} a. Show that f is a C00 function, and f(i) (0) = 0for all .
- 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:
- Let A= {x, y}: x
- Find the partial derivatives of f in terms of the derivatives of g and h if a. f(x,y) = g(x)h(y) b. f(x,y) = g(x) h(f) c. f(x,y) =g(x) d. f(x,y) =g(y) e. f(x,y) =g(x+y)
- If f: R2→R and D2f =0 and D2f =0, show that f is independent of the second variable. If D1f = D2f =0, show that f inconstant.
- Define g, h: {x€R2} |x| ≤ 1} →R by g(x,y) = (x,y, √1-x2-y2), h(x,y) = (x,y, - √1-x2-y2),
- Find expressions for the partial derivatives of the following functions: a. f(x,y) = f (g(x)k(y), g(x) + h(y) b. f(x,y,z) = f(g(+ y), h(y + z)) c. f(x,y,z) = f(xy, yz, zx) d. f(x,y) = f(x,g(x),
- 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) =
- Let f be defined as in Problem 2-4. Show that Dxf (0, 0) exists for all , but if g ≠ 0, , then Dx + yf (0,0) =Dxf (00 Dx + y f (0, 0) = Dx f (0 , 0) + Dyf (0, 0) Is not true for all x and
- 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).
- Show that the continuity of D1 f j at a may be eliminated from the hypothesis of Theorem 2-8.
- a. Let f : R → R be defined by F (x) = {x 2 sin 1/x) x ≠ 0, 0 x = 0.
- A function f: Rn → R is is homogeneous of degree m if f (tx) = tmf(x) for all x and t. If f is also differentiable, show that
- If f : Rn → R is differentiable and f (0) = 0, prove that there exist gi: Rn → R such that f (x) =
- Let A C Rn be an open set and f : A→ Rn a continuously differentiable 1-1 function such that det f1 (x) ≠ 0 for all . Show that f (A) is an open set and f -1: f (A) →A is
- Let A C Rn be an open set and f : A→ Rn a continuously differentiable 1-1 function such that det f1 (x) ≠ 0 for all . Show that f (A) is an open set and f -1: f (A) →A is differentiable.
- a. Let f: Rn → R be a continuously differentiable function. Show that f is not 1-1. b. Generalize this result to t the case of a continuously differentiable function f: Rn → Rm with m <
- a. If f : R → R satisfies f1 (a) ≠ 0 for all a €R, show that f is 1-1 on all of R.
- Use the function f : R → R defined by f (x) = { x/2 + x2 sin (1/x) x ≠ 0, 0 x = 0.
- Use the implicit function theorem to re-do Problem 2-15(c). Define f : R x Rn → Rn by
- Let f : R x R → R be differentiable. For each x € R defined gx: R → R By gx (y) = f(x,y). Suppose that for each x there is a unique y with gx1(y) =0; let c(x) be this y.
- Let f: [0, 1] x [0, 1] → R be defined by
- Let f: A→R be integrable and let g = except at finitely many points. Show that is integrable and f AF = f Ag.
- 1. Let f, g: A →R be integrable. a. For any partition of and any subrectangle of , show that and and therefore and .
- Let: A →R and P be a partition of A. Show that f is integrable if and only if for each subrectangle S the function f / s, which consists of f restricted to S, is integrable, and that in this
- Led f, g: A → R be integrable and suppose f < g. Show that fA f < f Ag.
- If f: A →R is integrable, show that |f| is integrable and |f A f| greater than f A |f|
- Let f: [0, 1] x [0, 1] → R be defined by Discuss.
- Prove that R = [a1, b1] x..x [an, bn] is not of content if ai < bi for i =1. n.
- a. Show that an unbounded set cannot have content 0.
- a. If C is a set of content 0, show that the boundary of C also has content 0. b. Give an example of a bounded set C of measure 0 such that the boundary of C does not have measure 0.
- Let A be the set of Problem 1-18. If T = ∑i = 1 (bi − ai)
- Let f: [a, b] be an increasing function. Show that {x: f is discontinuous at x} is a set of measure 0.
- a. Show that the set of all rectangles [a1, b1] x . x [an, bn] where each ai and each bi are rational can be arranged into a sequence (i.e. form a countable set). b. If A C Rn is any set and O is
- Show that if f, g: A → R are integrable, so is f ∙ g.
- Show that if C has content 0, then C C A for some closed rectangle A and C is Jordan-measurable and ∫ AXC = 0.
- Give an example of a bounded set C of measure 0 such that ∫ AXC does not exist.
- If C is a bounded set of measure 0 and ∫ AXC exists, show that ∫ AXC = 0.
- If f: A→R is non-negative and ∫ Af = 0, show that B = {x: f (x) ≠ 0} has measure 0.
- 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]
- 14. If is a closed rectangle, show that is Jordan measurable if and only if for every there is a partition of such that , where consists of all subrectangles intersecting and consists of
- If A is a Jordan measurable set and ε > 0, show that there is a compact Jordan measurable set C C A such that ∫ A − C1 < ε.
- Let C C A x B be a set of content 0. Let A1 C A be the set of all x Є A such that {y Є B: (x, y) Є C} is not of content 0. Show that A1 is a set of measure 0.
- Let C C [0, 1] x [0, 1] be the union of all {p/q} x [0, 1] where p/q is a rational number in [0, 1] written in lowest terms. Use C to show that the word ``measure" in Problem 3-23 cannot be replaced
- 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.
- Let f: [a, b] → R be integrable and non-negative, and let Af = {(x, y): a < x < b and 0 < x < f (x)}. Show that Af is Jordan measurable and has area ∫ ba f.
- If is continuous, show that
- Use Fubini's Theorem to give an easy proof that D1, 2f = D2, 1f if these are continuous.
- Use Fubini's Theorem to derive an expression for the volume of a set in R3 obtained by revolving a Jordan measurable set in the yz -plane about the -axis.
- If A = [a1, b1] x . x [an, bn] and f: A → R is continuous, define f: A → R by
- Let f: [a, b] x [c, d] → R be continuous and suppose D 2 f is continuous. Define f (y) = ∫ ba f (x, y) dx. Prove Leibnitz' Rule: f1 (y) = ∫ ba D2 f (x, y) dx.
- If f: [a, b] x [c, d] → R is continuous and D2f is continuous, define F (x, y) = ∫xa (t,y) dt a. Find D1F and D2F. (b) If G (x) = ∫ g(x) f (t, x) dt, find G1 (x).
- Let g1, g2: R2 → R be continuously differentiable and suppose D1 g2= D2 R1.. As in Problem 2-21, let
- a. Let g: Rn → Rn be a linear transformation of one of the following types:
- a. Suppose that f: (0, 1) → R is a non-negative continuous function. Show that ∫ (0, 1) exists if and only if lim Є→ ∫ c 1-c f exists. b. Let An = [1 - 1/2n, 1 - 1/2n
- Let An be a closed set contained in (n, n + 1). Suppose that f: R →R satisfies ∫Arf = (−1)n/n and f = 0 outside Un An.. Find two partitions of unity Φ and Ψ such that
- Use Theorem 3-14 to prove Theorem 3-13 without the assumption that g1 (x) ≠ 0.
- If g: Rn → Rn and detg1 (x) ≠ 0, prove that in some open set containing we can write g = to gn 0 ∙ ∙ ∙ o g1, 0.., where is of the form gi(x) = (x1, ∙ ∙
- If M is a k-dimensional manifold with boundary, prove that ∂M is a (k - 1) -dimensional manifold and M - ∂M is a k=dimensional manifold.
- Find a counter-example to Theorem 5-2 if condition (3) is omitted. Following the hint, consider f: (- 2π, 2π) →R2 defined by
- (a) Let A C Rn be an open set such that boundary A is an (n - 1) -dimensional manifold. Show that N = AU boundary A is an -dimensional manifold with boundary. (It is well to bear in mind the
- 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
- Prove that a k-dimensional (vector) subspace of Rn is a k-dimensional manifold.
- If f: Rn → Rn, the graph of f is {(x, y): y = f (x)}. Show that the graph of is an -dimensional manifold if and only if is differentiable.
- 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
- a. If M is a k-dimensional manifold in Rn and k < n, show that M has measure 0. b. If M is a closed -dimensional manifold with boundary in Rn, show that the boundary of M is ∂M. Give a
- Show that Mx consists of the tangent vectors at t of curves in M with c (t) = x.
- Suppose C is a collection of coordinate systems for M such that (1) For each x Є M there is f Є C which is a coordinate system around ; (2) if f, g Є C, then det (f -1 0 g) 2 > 0.
- If M is an -dimensional manifold-with-boundary in Rn, define μ as the usual orientation of M x = Rnx (the orientation μ so defined is the usual orientation of M. If xЄ∂M, show
- a. If f is a differentiable vector field on M C Rn, show that there is an open set AЭM and a differentiable vector field F on A with F(x) = F (x) for xЄM. b. If M is closed, show that
- Let f: A → Rp be as in Theorem 5-1. a. If x Є M = g-1(0), let h: U → Rn be the essentially unique diffeomorphism such that goh (y) = (y n – p + 1 . yn) and h (0) = x. Define f:
- If M C R n is an orientable (n - 1)-dimensional manifold, show that there is an open set A C Rn and a differentiable g: A→ R1 so that M = g-1 (0) and g1 (x) has rank 1 for x ЄM.
- 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
- 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
- a. Let Ί: Rn → Rn be self-adjoint with matrix A = (aij), so that aij = aji. If f (x) = =Σ aij xixj, show that Dkf (x) = 2 Σj = 1 akjxj. By considering the maximum of on Sn-1
- If M is an -dimensional manifold (or manifold-with-boundary) in R n, with the usual orientation, show that ∫ fdx1 ^ . ^ dx n, as defined in this section, is the same as ∫ M f, as defined
- a. Show that Theorem 5-5 is false if M is not required to be compact. b. Show that Theorem 5-5 holds for noncom-pact M provided that w vanishes outside of a compact subset of M.
- If w is a (k- 1) -form on a compact k-dimensional manifold M, prove that ∫Mdw =0. Give a counter-example if M is not compact.
- An absolute k-tensor on v is a function Vk →R of the form |w| for w Є Ak (V). An absolute k-form on M is a function such that n (x) is an absolute k-tensor on Mx. Show that ∫Mn can
- If M1CRN is an -dimensional manifold-with-boundary and M 2 C M1 - ∂M1 is an -dimensional manifold with boundary, and M1, M2 are compact, prove that
- If M is an oriented one-dimensional manifold in RN and c: [0, 1] →M is orientation-preserving, show that