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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 and only if there is a function f: R→R such that f(x, y) = g(x). What is f1 (a, b) in terms of g1?
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 ≠0,
0 x = 0.
(a) If x € R2 and h: R →R is defined by h (t) = f (tx) show that h is differentiable.
(b). show that f is not differentiable at (0, 0) unless g = 0.
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 the form g(x) = a0 + a1 (x – a ) such that f and g and g are equal up to first order at a.
(b). if f1 (a),., f(n) (a) exist, show that f and the function g defined by
g(x) = f(i) (a)/ i! (x – a)i
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) =xy+z
f. f(x, y, z) =(x + y)z
g. f(x, y) = sin (xy)
h. f(x, y) = sin (xy) cos(3)
i. f(x, y) =(sin(xy), sin (xsin (y)),xy)
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, y1) + f(x, y2)
(a) Prove that if f is bilinear, then
(b) Prove that Df (a, b) (x, y) = f (a,y) + f(x,b).
(c) (Show that the formula for Dp (a, b) in theorem 2-3 is a special case of (b
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) = .
(c) If f: R → Rn is differentiable and |f(t) = 1 for all t, show that = 0.
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 by g(x) = f(x1,.,xi-1, x,xi +1, ., xk) is a linear transformation.
(a) If is multi linear and i ≠ j, show that for (h=(h1, ., hk), with hi € Ei, we have
Prove that
(b) Df (a1,., ak) x1, ., xk) = =1 f(a1,., ai-1, xi, ai+1,., ak)
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
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 sin(z)))
e. f(x,y,z)=xy2
f. f(x,y,z)=xy=z
g. f(x,y,z)=(x +y)2
h. f(x,y)= sin(xy)
i. f(x,y)= (sin (xy)) cos(3)
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: R2→R such that D1f (x,y)=x and D1f (x,y)=y
Let A= {x, y}: x <0, or x ≥ 0 and y ≠ 0}.
a. If f: A→R and D1f =D2f = 0, show that f is a constant.
b. Find a function f: A→R such that D2f =0 but f is not independent of the second variable.
Find the partial derivatives of f in terms of the derivatives of g and h if
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), h(x,y))
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)..
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 all y.
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 differentiable. Show also that f (b) is open for any open set B C A.
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. Show also that f (b) is open for any open set B C A.
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 < n.
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 case f A f = ∑S f S f | S
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|
Let f: [0, 1] x [0, 1] → R be defined by
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) <, show that the boundary of A does not have measure 0.
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 an open cover of A, show that there is a sequence U1, U2, U3,. of members of O which also cover A.
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 allsubrectangles contained in .
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 with ``content".
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 +1] Suppose that f: (0, 1) →R satisfies ∫Arf = (-1)n/n and f(x) = 0 for all x Є Un An. Show that ∫(0,1)f does not exist, but limЄ→∫(Є, 1 - Є)f = log 2.
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 ∑ǿЄΦ∫Rǿ∙f and ∑ǿЄΦ∫Rψ∙f converge absolutely to different values.
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, ∙ ∙ ∙ Fi (x) , ∙ ∙ ∙ Xn), and T is a linear transformation. Show that we can write g = gn o ∙ ∙ ∙ 0g1 if and only if g1 (x) is a diagonal matrix.
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 following example: if A = {x ЄRn}: |x| < 1 or 1 < |x| < 2}, then N = AU boundary A is a manifold with boundary, but ∂ N ≠ boundary A.
(b) Prove a similar assertion for an open subset of an n-dimensional manifold.
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.
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 manifold. Example: the tours (Figure 5-4).
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 counter-example if M is not closed.
c. If M is a compact -dimensional manifold with boundary in Rn, show that M is Jordan-measurable.
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. Show that there is a unique orientation of M such that is orientation-preserving for all f Є C.
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 that the two definitions of n (x) given above agree.
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 we can choose A = Rn.
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: Rn- p → f: Rn-p →Rn by f (a) = h (0, a). Show that is 1-1 so that the vectors f * ((e1)0). f* ((en-p)0) are linearly independent.
b. Show that the orientations μ can be defined consistently, so that M is orientable.
c. If P = 1, show that the components of the outward normal at are some multiple of D1g (x). . . Dng(x).
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 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?
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 show that there is xЄSn-1 and ^ ЄR with Tx = ^x.
b. If V = {yЄRn: = 0}, show that Ί(v) CV and Ί: V and Ί: V → V is self-adjoint.
c. Show that Ί has a basis of eigenvectors.
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 in Chapter 3.
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 be defined, even if M is not orientable.
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
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