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数学代写-MAT257

时间：2021-01-16

MAT257 Term Test 2 Rejects

The following questions were a part of a question pool for the 2020-21 MAT257 Term Test 2,

but at the end, they were not included.

1. Let A be a rectangle in Rn and let f , g : A → R, where f is integrable on A and g is equal to

f except on finitely many points. Show from basic definitions that g is also integrable on A

and that

∫

A

f =

∫

A

g.

Tip. “From basic definitions” means “not using any of the theorems that came after the definitions

that are necessary to make the question meaningful”. In our case those definitions are those of lower

and upper sums, integrability, and the integral. Yet words like “measure-0”, whether or not they are

relevant, are forbidden.

2. (a) Show that the boundary of a set of content-0 is also of content-0.

(b) Give an example of a set of measure-0 whose boundary is not of measure-0.

3. Show that if f : A → R is integrable on a rectangle A, and if g : R → R is continuous and

bounded on f (A), then g ◦ f is also integrable on A.

4. Show that if a set A ⊂ Rn is Jordan measurable, then there is a finite collection R of nearly

disjoint rectangles in Rn (meaning, disjoint except perhaps for their boundaries) and a sub-

collection R′ of R, such that

(1)

⋃

R∈R′

R ⊂ A ⊂

⋃

R∈R

R and

∑

R∈R\R′

v(R) <

1

257

.

5. (a) Show that if a set B is bounded, has measure 0, and its characteristic function χB is

integrable, then

∫

χB = 0.

(b) Give an example of a bounded measure-0 set whose characteristic function is not inte-

grable.

6. Show that if a non-negative continuous function f defined on some rectangle A in Rn has

integral equal to 0, then f is identically equal to 0.

7. Prove Young’s inequality: if f is a continuous strictly increasing function on Rwith f (0) = 0

and if a and b are non-negative numbers, then∫ a

0

f (x)dx +

∫ b

0

f −1(y)dy ≥ ab.

Hint. Draw the graph of f and try to interpret the two integrals and the product ab as areas.

8. Suppose A and B are two Jordan measurable subsets of Rn that have the property that for

every t ∈ R, the (n − 1)-dimensional volume of the slice of A at height t (meaning, of

{x ∈ Rn−1 : (x, t) ∈ A}) is equal to the (n − 1)-dimensional volume of the slice of B at height

t. Prove that v(A) = v(B).

1/2

9. Prove that if f : Rn → Rn is smooth and has an invertible differential at 0, then near 0 it

can be written as a composition g1 ◦ g2 ◦ · · · ◦ gn ◦ T , where T is a “permutation map” that

merely permutes the coordinates of x = (x1 x2 . . . xn) ∈ Rn, and each gi changes the value

of only one coordinate; precisely, gi(x1 x2 . . . xn) = (x1 . . . xi−1 hi x j+1 . . . xn), where the

hi : Rn → R are smooth.

10. Let f be a possibly-unbounded function defined on a possibly-unbounded open subset A of

Rn, and assume that f is integrable on A. Let B be an open subset of A. Prove that f is

integrable also on B.

11. Compute the volume of the ellipsoid {(x, y, z) : 2x2 + 3y2 + 5z2 ≤ 1}. You may use the fact

that the volume of the ball of radius R in R3 is 43piR

3.

12. Let f : R2x,y → R be a function that has continuous second derivatives, and let A be a rectangle

in R2x,y

(a) Use Fubini to show that

∫

R

∂x(∂y f ) =

∫

R

∂y(∂x f ).

(b) Use the above result to show that ∂x(∂y f ) = ∂y(∂x f ).

13. Let Tθ be the “rotation by θ” matrix

(

cos θ − sin θ

sin θ cos θ

)

, and let B be a Jordan-measurable subset

of R2. Prove that v(B) = v(TθB).

14. Let A be a subset of Rn, and letU andV be open covers of A. Show that if {φi} is a partition

of unity for A subordinate toU and {ψ j} is a partition of unity for A subordinate toV, then

{φiψ j} is a partition of unity for A subordinate toW B {U ∩ V : U ∈ U, V ∈ V}.

2/2

The following questions were a part of a question pool for the 2020-21 MAT257 Term Test 2,

but at the end, they were not included.

1. Let A be a rectangle in Rn and let f , g : A → R, where f is integrable on A and g is equal to

f except on finitely many points. Show from basic definitions that g is also integrable on A

and that

∫

A

f =

∫

A

g.

Tip. “From basic definitions” means “not using any of the theorems that came after the definitions

that are necessary to make the question meaningful”. In our case those definitions are those of lower

and upper sums, integrability, and the integral. Yet words like “measure-0”, whether or not they are

relevant, are forbidden.

2. (a) Show that the boundary of a set of content-0 is also of content-0.

(b) Give an example of a set of measure-0 whose boundary is not of measure-0.

3. Show that if f : A → R is integrable on a rectangle A, and if g : R → R is continuous and

bounded on f (A), then g ◦ f is also integrable on A.

4. Show that if a set A ⊂ Rn is Jordan measurable, then there is a finite collection R of nearly

disjoint rectangles in Rn (meaning, disjoint except perhaps for their boundaries) and a sub-

collection R′ of R, such that

(1)

⋃

R∈R′

R ⊂ A ⊂

⋃

R∈R

R and

∑

R∈R\R′

v(R) <

1

257

.

5. (a) Show that if a set B is bounded, has measure 0, and its characteristic function χB is

integrable, then

∫

χB = 0.

(b) Give an example of a bounded measure-0 set whose characteristic function is not inte-

grable.

6. Show that if a non-negative continuous function f defined on some rectangle A in Rn has

integral equal to 0, then f is identically equal to 0.

7. Prove Young’s inequality: if f is a continuous strictly increasing function on Rwith f (0) = 0

and if a and b are non-negative numbers, then∫ a

0

f (x)dx +

∫ b

0

f −1(y)dy ≥ ab.

Hint. Draw the graph of f and try to interpret the two integrals and the product ab as areas.

8. Suppose A and B are two Jordan measurable subsets of Rn that have the property that for

every t ∈ R, the (n − 1)-dimensional volume of the slice of A at height t (meaning, of

{x ∈ Rn−1 : (x, t) ∈ A}) is equal to the (n − 1)-dimensional volume of the slice of B at height

t. Prove that v(A) = v(B).

1/2

9. Prove that if f : Rn → Rn is smooth and has an invertible differential at 0, then near 0 it

can be written as a composition g1 ◦ g2 ◦ · · · ◦ gn ◦ T , where T is a “permutation map” that

merely permutes the coordinates of x = (x1 x2 . . . xn) ∈ Rn, and each gi changes the value

of only one coordinate; precisely, gi(x1 x2 . . . xn) = (x1 . . . xi−1 hi x j+1 . . . xn), where the

hi : Rn → R are smooth.

10. Let f be a possibly-unbounded function defined on a possibly-unbounded open subset A of

Rn, and assume that f is integrable on A. Let B be an open subset of A. Prove that f is

integrable also on B.

11. Compute the volume of the ellipsoid {(x, y, z) : 2x2 + 3y2 + 5z2 ≤ 1}. You may use the fact

that the volume of the ball of radius R in R3 is 43piR

3.

12. Let f : R2x,y → R be a function that has continuous second derivatives, and let A be a rectangle

in R2x,y

(a) Use Fubini to show that

∫

R

∂x(∂y f ) =

∫

R

∂y(∂x f ).

(b) Use the above result to show that ∂x(∂y f ) = ∂y(∂x f ).

13. Let Tθ be the “rotation by θ” matrix

(

cos θ − sin θ

sin θ cos θ

)

, and let B be a Jordan-measurable subset

of R2. Prove that v(B) = v(TθB).

14. Let A be a subset of Rn, and letU andV be open covers of A. Show that if {φi} is a partition

of unity for A subordinate toU and {ψ j} is a partition of unity for A subordinate toV, then

{φiψ j} is a partition of unity for A subordinate toW B {U ∩ V : U ∈ U, V ∈ V}.

2/2