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证明代写-MAT237
时间：2022-06-20
Problems
1. Let a, b, c 2 R and let Sa,b,c = {(x , y, z) : ax2 + b y2 + cz2 = 0, x2 + y2 + z2 = 1}.
(1a) For which values a, b, c is Sa,b,c a regular surface of dimension 2?
MAT237 Problem Set 4 - Page 2 of 11 June 20, 2022
(1b) For which values a, b, c is Sa,b,c a regular curve?
MAT237 Problem Set 4 - Page 3 of 11 June 20, 2022
(1c) For which values a, b, c is Sa,b,c discrete? (A set S ✓ Rn is discrete if for each x 2 S, there exists " > 0
such that B"(x)\ S = {x}.)
(1d) For which values a, b, c is Sa,b,c empty?
MAT237 Problem Set 4 - Page 4 of 11 June 20, 2022
2. For a fixed A2 R, define the curve CA in R2 by the equation
y2 2x y = x4 + Ax + 4.
You can view this family of curves on Desmos. Note this demo will not help you justify your answers below.
(2a) Prove that if A 6= 6 and A 6= 6, then CA is a regular curve. You may use WolframAlpha to solve a
1-variable quartic equation.
MAT237 Problem Set 4 - Page 5 of 11 June 20, 2022
(2b) This family of curves can instead be viewed as the z-slices of the surface S in R3 defined by the equation
y2 2x y = x4 + xz + 4
Use the implicit function theorem to show that this equation defines a 2-dimensional regular surface
(2c) The curves C6 and C6 are regular at each point except (1,1) and (1,1) respectively. These points
where C6 and C6 fail to be regular are called singularities. View this surface and family of curves on
Math3d. (Google Chrome is the most stable browser for Math3D.) Informally explain how the shape of
the surface appears to relate to the singularities of C6 and C6. Your explanation must discuss both the
surface and the curves.
MAT237 Problem Set 4 - Page 6 of 11 June 20, 2022
3. Let U ,V ⇢ Rn be open.
(3a) Suppose U 0 ⇢ U is open and a 2 U 0. Let g : U ! V . Use the " definition of the limit to prove that
@ g(a)
@ xi
exists if and only if @ g|U0 (a)@ xi exists. Furthermore, prove that
@ g|U0 (a)
@ xi
= @ g(a)@ xi if either exist.
MAT237 Problem Set 4 - Page 7 of 11 June 20, 2022
(3b) Let g : U ! V . Use part (a) to prove that g is a diffeomorphism if and only if g is bijective and a local
diffeomorphism at each point in U .
(3c) Conclude using part (b) that if g : U ! V is a C1 bijection such that Dg(u) is invertible for all u 2 U ,
then g is a diffeomorphism.
MAT237 Problem Set 4 - Page 8 of 11 June 20, 2022
4. Find the maximum value for f (x , y, z) = x3 2x y + y2 + 2z on S = {(x , y, z) 2 R3 : 0 z 1 x2 y2}.
You must fully justify your solution. You may not use WolframAlpha to solve any systems of equations that
arise in your solution.
MAT237 Problem Set 4 - Page 9 of 11 June 20, 2022
5. A common use for Lagrange multipliers is to maximize profit subject to a budget constraint. How can a
company make the most money with what they currently have to spend? In that context, the multiplier
has a very concrete interpretation: is the marginal maximum profit with respect to the constraint. More
precisely, if M is the maximum profit obtained with budget $c and is the corresponding multiplier from the
Lagrange multiplier system, the
dM
dC
= . We will investigate this surprising fact.
Let f , g : Rn ! R be C1. Suppose f attains a unique maximum on Sc = {x 2 Rn : g(x) = c} for each c 2 R.
Define M(c) = max{ f (x) 2 R : x 2 Rn, g(x) = c}. Let x⇤(c) be the location of the maximum in Sc and let
⇤(c) be the multiplier in the Lagrange system corresponding to x⇤(c). You may assume x⇤ and ⇤ are C1
functions in c.
(5a) The Lagrangian function corresponding to this situation is L(x ,, c) = f (x) (g(x) c). Write M(c)
as a composition of functions using L. No justification is required.
(5b) Prove that for a fixed C 2 R, (x⇤(C),⇤(C)) is a critical point of L(x ,,C).
MAT237 Problem Set 4 - Page 10 of 11 June 20, 2022
(5c) Prove that
dM(c)
dc
= ⇤(c).
MAT237 Problem Set 4 - Page 11 of 11 June 20, 2022
1. Let a, b, c 2 R and let Sa,b,c = {(x , y, z) : ax2 + b y2 + cz2 = 0, x2 + y2 + z2 = 1}.
(1a) For which values a, b, c is Sa,b,c a regular surface of dimension 2?
MAT237 Problem Set 4 - Page 2 of 11 June 20, 2022
(1b) For which values a, b, c is Sa,b,c a regular curve?
MAT237 Problem Set 4 - Page 3 of 11 June 20, 2022
(1c) For which values a, b, c is Sa,b,c discrete? (A set S ✓ Rn is discrete if for each x 2 S, there exists " > 0
such that B"(x)\ S = {x}.)
(1d) For which values a, b, c is Sa,b,c empty?
MAT237 Problem Set 4 - Page 4 of 11 June 20, 2022
2. For a fixed A2 R, define the curve CA in R2 by the equation
y2 2x y = x4 + Ax + 4.
You can view this family of curves on Desmos. Note this demo will not help you justify your answers below.
(2a) Prove that if A 6= 6 and A 6= 6, then CA is a regular curve. You may use WolframAlpha to solve a
1-variable quartic equation.
MAT237 Problem Set 4 - Page 5 of 11 June 20, 2022
(2b) This family of curves can instead be viewed as the z-slices of the surface S in R3 defined by the equation
y2 2x y = x4 + xz + 4
Use the implicit function theorem to show that this equation defines a 2-dimensional regular surface
(2c) The curves C6 and C6 are regular at each point except (1,1) and (1,1) respectively. These points
where C6 and C6 fail to be regular are called singularities. View this surface and family of curves on
Math3d. (Google Chrome is the most stable browser for Math3D.) Informally explain how the shape of
the surface appears to relate to the singularities of C6 and C6. Your explanation must discuss both the
surface and the curves.
MAT237 Problem Set 4 - Page 6 of 11 June 20, 2022
3. Let U ,V ⇢ Rn be open.
(3a) Suppose U 0 ⇢ U is open and a 2 U 0. Let g : U ! V . Use the " definition of the limit to prove that
@ g(a)
@ xi
exists if and only if @ g|U0 (a)@ xi exists. Furthermore, prove that
@ g|U0 (a)
@ xi
= @ g(a)@ xi if either exist.
MAT237 Problem Set 4 - Page 7 of 11 June 20, 2022
(3b) Let g : U ! V . Use part (a) to prove that g is a diffeomorphism if and only if g is bijective and a local
diffeomorphism at each point in U .
(3c) Conclude using part (b) that if g : U ! V is a C1 bijection such that Dg(u) is invertible for all u 2 U ,
then g is a diffeomorphism.
MAT237 Problem Set 4 - Page 8 of 11 June 20, 2022
4. Find the maximum value for f (x , y, z) = x3 2x y + y2 + 2z on S = {(x , y, z) 2 R3 : 0 z 1 x2 y2}.
You must fully justify your solution. You may not use WolframAlpha to solve any systems of equations that
arise in your solution.
MAT237 Problem Set 4 - Page 9 of 11 June 20, 2022
5. A common use for Lagrange multipliers is to maximize profit subject to a budget constraint. How can a
company make the most money with what they currently have to spend? In that context, the multiplier
has a very concrete interpretation: is the marginal maximum profit with respect to the constraint. More
precisely, if M is the maximum profit obtained with budget $c and is the corresponding multiplier from the
Lagrange multiplier system, the
dM
dC
= . We will investigate this surprising fact.
Let f , g : Rn ! R be C1. Suppose f attains a unique maximum on Sc = {x 2 Rn : g(x) = c} for each c 2 R.
Define M(c) = max{ f (x) 2 R : x 2 Rn, g(x) = c}. Let x⇤(c) be the location of the maximum in Sc and let
⇤(c) be the multiplier in the Lagrange system corresponding to x⇤(c). You may assume x⇤ and ⇤ are C1
functions in c.
(5a) The Lagrangian function corresponding to this situation is L(x ,, c) = f (x) (g(x) c). Write M(c)
as a composition of functions using L. No justification is required.
(5b) Prove that for a fixed C 2 R, (x⇤(C),⇤(C)) is a critical point of L(x ,,C).
MAT237 Problem Set 4 - Page 10 of 11 June 20, 2022
(5c) Prove that
dM(c)
dc
= ⇤(c).
MAT237 Problem Set 4 - Page 11 of 11 June 20, 2022
