A. Let f (x, y) be a real-valued function. Show that if f x (x, y) and f y (x, y) are both continuous at (0, 0), then f (x, y) is differentiable at (0, 0).

(1)

March 10, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A. Let f (x, y) be a real-valued function. Show that if f _x (x, y) and f _y (x, y) are both continuous at (0, 0), then f (x, y) is differentiable at (0, 0).

B. Consider the function f (x, y) =



 

  xy

px ² + y ² if (x, y) 6= (0, 0) 0 if (x, y) = (0, 0) .

(a) Show that f is continuous at (0, 0).

(b) Show that f _x (0, 0) and f _y (0, 0) both exist and evaluate their values.

(c) Show that f is not differentiable at (0, 0).

C. Consider the function f (x, y) =







x ² tan ⁻¹ (y/x) − y ² tan ⁻¹ (x/y) if x 6= 0 and y 6= 0

0 otherwise

.

(a) Evaluate f _x (x, y) and f _y (x, y) for x 6= 0 and y 6= 0.

(b) Evaluate f _x (0, 0) and f _y (0, 0), and show that f is differentiable at (0, 0). What is the tangent plane of the surface z = f (x, y) at (0, 0, 0)?

(c) Evaluate f _x (0, y) for y 6= 0 and f _y (x, 0) for x 6= 0. Show that f _xy (0, 0) 6= f _yx (0, 0).

D. Let f (x, y), u(x, y) and v(x, y) be C ² functions. Suppose that f _xx + f _yy = 0 and u _x = v _y , u _y = −v _x , show that the function φ(x, y) = f (u(x, y), v(x, y)) also satisfies φ xx + φ yy = 0.

1

(2)

March 24, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A. Let u(x, y) : R ² → R be a C ² function. Express u _xx + u _yy in polar coordinate.

B. Let f (x, y) : R ² → R be a C ¹ function. Show that d dy

Z b a

f (x, y)dx = Z b

a

∂

∂y f (x, y)dx.

Hint. Every continuous function defined on a bounded and closed set D ⊂ R ⁿ is uniformly continuous.

C. Consider the line integral Z

Γ

L, where L = Adx + Bdy + Cdz defined on R ³ . Prove that L is exact, that is, L = df for some f on R ³ , if and only if the integral is independent to the path, which means that it only depends on the end points of Γ.

D. (a) Evaluate Z

zdx + xdy + ydz over the arc of the helix



 

 

 

 

x = cos t y = sin t z = t

from (1, 0, 0) to (1, 0, 2π).

(b) Evaluate

Z ydx + xdy

1 + x ² y ² over the arc of y = sin 1

x from (1/2π, 0) to (1/π, 0).

Hint. Check if the differential form is exact.

1

(3)

April 7, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A. Let F (x, y) : R ² → R be a C ¹ function. Suppose that F (x ₀ , y ₀ ) = 0 and F _y (x ₀ , y ₀ ) > 0, show that there exists a δ > 0 and a unique C ¹ function f (x) : (x 0 − δ, x 0 + δ) → R s.t. F (x, f (x)) = 0 for all x ∈ (x ₀ − δ, x ₀ + δ). That is, y can be solved uniquely as a C ¹ function of x near x ₀ .

B. Let F (x, y, z) = x + y + z − sin xy − sin yz − sin xz.

(a) Show that F (x, y, z) = 0 can be solve for z = f (x, y), where f is a C ¹ function, near (0, 0, 0).

(b) Find f _x (0, 0) and f _y (0, 0). What is the tangent plane of the surface F (x, y, z) = 0 at (0, 0, 0)?

C. Find the stationary points of f (x, y) = x ³ + (y − x)(2y + x) − 3

2 y ² and determine whether they are local maximum, local minimum or saddle point.

D. Find the condition that the quadrilateral with given edges a, b, c, d includes the greatest area. And find its area.

Hint. Suppose the pairs a, b and c, d are adjacent. Let φ be the angle between a and b, ψ that between c and d. Express the area as a function of φ and ψ, and use cosine law to construct a constrain for φ and ψ.

1

(4)

April 28, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A. Consider a circle C, which lies in the xz-plane, with center (a, 0, 0) and radius r < |a|. Let Γ be the torus obtained by rotating C about z-axis. Find the tangent plane of Γ at point

a

√ 2 + r 2 , a

√ 2 + r 2 , r

√ 2

. B. Calculate the first fundamental form of the surface of revolution given by r = p

x ² + y ² = f (z), where f is a C ¹ function, in terms of the cylindrical coordinates z and θ = tan ⁻¹ y

x . C. Let S be the sphere x ² + y ² + z ² = 1.

(a) Use stereographic projection from the north pole (0, 0, 1) to the plane z = 0 to obtain a parametric representation for S\{(0, 0, 1)}.

(b) Show that the parametrization r(u, v) : R ² → R ³ in (a) is conformal. That is, if two curves on z = 0, which intersect at (u, v, 0), are orthogonal at (u, v, 0), then the two image curves on the sphere are also orthogonal at r(u, v).

D. Consider the function U = F(X) = (x ² − y ² , xy).

(a) Obtain an iterative approximation G(X), which depends on given U, for the inverse transfor- mation F ⁻¹ (U) near X ₀ = (1, 1) or U ₀ = (0, 1). Verify that the fixed point X _fixed of G satisfies U = F(X _fixed ).

(b) Show that there exists a δ > 0 s.t. for any U ∈ B _δ (U ₀ ) the iteration X _n+1 = G(X _n ) with initial value X ₀ converges to a limit, denoted by X(U).

1

(5)

May 12, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A1. Show that S is Jordan measurable in R ⁿ if and only if ∂S has Jordan measure 0 in R ⁿ .

A2. Let S and T be any disjoint sets whose union has area. Show that A ⁺ (S) + A ⁻ (T ) = A(S ∪ T ).

B. (a) Find the volume common to the two cylinders x ² + z ² ≤ 1 and y ² + z ² ≤ 1.

(b) Find the volume common to the three cylinders x ² + z ² ≤ 1, y ² + z ² ≤ 1 and x ² + y ² ≤ 1.

C. (a) Evaluate the integral Z 1

0 Z 1 y

e ^x

²

dxdy.

(b) Evaluate the integral Z 1

0 Z

√ 1−z

²

0 Z

√

1−y

²

−z

²

0 (x ² + y ² + z ² )xyzdxdydz.

D1. For a, b, c ∈ R, evaluate the integral Z

{x

²

+y

²

+z

²

≤1}

cos(ax + by + cz)dxdydz.

D2. Evaluate the integral Z

{x

²

+y

²

+z

²

≤1}

e ^x+y+z dxdydz.

Hint. Consider the change of variable



 

 

 

 

x ⁰ = (x + y + z)/ √ 3 y ⁰ = (x − z)/ √

2 z ⁰ = (x − 2y + z)/ √

6 .

1

(6)

May 26, 2011

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A. Determine whether the improper integral Z Z Z

R

³

dxdydz

(1 + x ² + y ² + z ² ) ² converges or diverges. If it con- verges, evaluate the integral.

B. Find the area of surface r(r, θ) = (r cos θ, r sin θ, θ) with 0 ≤ r ≤ 1 and 0 ≤ θ ≤ 2π.

C. Find the volume of the n-simplex described by x k ≥ 0 for k = 1, 2, · · · , n and x ₁ a ₁ + x ₂

a ₂ + · · · + x _n a _n ≤ 1.

D. (a) Evaluate the improper integral Z ∞

0 e ^−tx cos x dx for any fixed t.

(b) Show that the integral in (a) converges uniformly in t.

(c) Evaluate the integral Z ∞

0 e ^−bx − e ^−ax

x cos x dx.

E. Assume we have known that the surface area of z = f (x, y) in the region (x, y) ∈ R is Z

R

q

1 + f _x ² + f _y ² dxdy

Now, there is a surface in the parametric form x = φ(u, v), y = ψ(u, v), z = χ(u, v)

Show that if we consider a portion R ⁰ of u, v-plane where the Jacobian d(x, y)

d(u, v) is not zero, then the surface area of the image of R ⁰ is

Z

R

⁰

√

EG − F ² dudv

Where E, F , G are the elements of the first fundamental form.

F. Let the region R be an unbounded set. Assume that we can find {R _n } s.t. R _n ⊂ R _n+1 ⊂ R in each of where f (x, y) is continuous. Assume we have

S ⊂ R _n for some n, Z

R

n

|f (x, y)|dA < u,

for every bounded, closed set S ⊂ R and u is independent to n, show that I = lim

n→∞

Z

R

n

f (x, y)dA

exists and is independent to the choice of approximating sequence R _n .

1

(7)

June 9, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A. Let L = −ydx + xdy x ² + y ² . (a) Evaluate

Z

C

ε

L, where C _ε is a circle with center (0, 0) and radius ε.

(b) Show that for any piecewise C ¹ simple closed curve C, Z

C

L =







2π if (0,0) lies in the close curve

0 if (0,0) does not lie in the close curve .

B. Use Green’s Theorem to prove the change of variable formula Z Z

R

f (x, y)dxdy = Z Z

S

f (x(u, v), y(u, v)) ∂(x, y)

∂(u, v) dudv,

provided that the transformation (u, v) 7→ (x(u, v), y(u, v)) from S to R is one-to-one and C ² with positive Jacobian.

C. Calculate Z Z

S

zdx ∧ dy − xdy ∧ dz, where S is the spherical cap x ² + y ² + z ² = 1, x > 1/2, oriented positively with respect to the normal pointing to infinity.

D. Let F(x, y, z) = (0, 0, z) and S be the helicoid r(u, v) = (r cos θ, r sin θ, θ), 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π.

Evaluate the flux Z Z

S

F · ndS, where n is the upward normal of S.

1

A. Let f (x, y) be a real-valued function. Show that if f x (x, y) and f y (x, y) are both continuous at (0, 0), then f (x, y) is differentiable at (0, 0).

March 10, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A. Let f (x, y) be a real-valued function. Show that if f x (x, y) and f y (x, y) are both continuous at (0, 0), then f (x, y) is differentiable at (0, 0).

B. Consider the function f (x, y) =



 

  xy

px 2 + y 2 if (x, y) 6= (0, 0) 0 if (x, y) = (0, 0) .

(a) Show that f is continuous at (0, 0).

(b) Show that f x (0, 0) and f y (0, 0) both exist and evaluate their values.

(c) Show that f is not differentiable at (0, 0).

C. Consider the function f (x, y) =







x 2 tan −1 (y/x) − y 2 tan −1 (x/y) if x 6= 0 and y 6= 0

0 otherwise

.

(a) Evaluate f x (x, y) and f y (x, y) for x 6= 0 and y 6= 0.

(b) Evaluate f x (0, 0) and f y (0, 0), and show that f is differentiable at (0, 0). What is the tangent plane of the surface z = f (x, y) at (0, 0, 0)?

(c) Evaluate f x (0, y) for y 6= 0 and f y (x, 0) for x 6= 0. Show that f xy (0, 0) 6= f yx (0, 0).

D. Let f (x, y), u(x, y) and v(x, y) be C 2 functions. Suppose that f xx + f yy = 0 and u x = v y , u y = −v x , show that the function φ(x, y) = f (u(x, y), v(x, y)) also satisfies φ xx + φ yy = 0.

1

March 24, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A. Let u(x, y) : R 2 → R be a C 2 function. Express u xx + u yy in polar coordinate.

B. Let f (x, y) : R 2 → R be a C 1 function. Show that d dy

Z b a

f (x, y)dx = Z b

a

∂

∂y f (x, y)dx.

Hint. Every continuous function defined on a bounded and closed set D ⊂ R n is uniformly continuous.

C. Consider the line integral Z

Γ

L, where L = Adx + Bdy + Cdz defined on R 3 . Prove that L is exact, that is, L = df for some f on R 3 , if and only if the integral is independent to the path, which means that it only depends on the end points of Γ.

D. (a) Evaluate Z

zdx + xdy + ydz over the arc of the helix



 

 

 

 

x = cos t y = sin t z = t

from (1, 0, 0) to (1, 0, 2π).

(b) Evaluate

Z ydx + xdy

1 + x 2 y 2 over the arc of y = sin 1

x from (1/2π, 0) to (1/π, 0).

Hint. Check if the differential form is exact.

1

April 7, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

B. Let F (x, y, z) = x + y + z − sin xy − sin yz − sin xz.

(a) Show that F (x, y, z) = 0 can be solve for z = f (x, y), where f is a C 1 function, near (0, 0, 0).

(b) Find f x (0, 0) and f y (0, 0). What is the tangent plane of the surface F (x, y, z) = 0 at (0, 0, 0)?

C. Find the stationary points of f (x, y) = x 3 + (y − x)(2y + x) − 3

2 y 2 and determine whether they are local maximum, local minimum or saddle point.

D. Find the condition that the quadrilateral with given edges a, b, c, d includes the greatest area. And find its area.

Hint. Suppose the pairs a, b and c, d are adjacent. Let φ be the angle between a and b, ψ that between c and d. Express the area as a function of φ and ψ, and use cosine law to construct a constrain for φ and ψ.

1

April 28, 2010

Dept. ID No. Name:

Make sure to give sufficient reason in each problem or you will NOT get any credit for your answer.

A. Consider a circle C, which lies in the xz-plane, with center (a, 0, 0) and radius r < |a|. Let Γ be the torus obtained by rotating C about z-axis. Find the tangent plane of Γ at point

 a

√ 2 + r 2 , a

√ 2 + r 2 , r

√ 2

 . B. Calculate the first fundamental form of the surface of revolution given by r = p

x 2 + y 2 = f (z), where f is a C 1 function, in terms of the cylindrical coordinates z and θ = tan −1 y

x . C. Let S be the sphere x 2 + y 2 + z 2 = 1.

(a) Use stereographic projection from the north pole (0, 0, 1) to the plane z = 0 to obtain a parametric representation for S\{(0, 0, 1)}.

(b) Show that the parametrization r(u, v) : R 2 → R 3 in (a) is conformal. That is, if two curves on z = 0, which intersect at (u, v, 0), are orthogonal at (u, v, 0), then the two image curves on the sphere are also orthogonal at r(u, v).

D. Consider the function U = F(X) = (x 2 − y 2 , xy).

(a) Obtain an iterative approximation G(X), which depends on given U, for the inverse transfor- mation F −1 (U) near X 0 = (1, 1) or U 0 = (0, 1). Verify that the fixed point X fixed of G satisfies U = F(X fixed ).

A. Let f (x, y) be a real-valued function. Show that if f _x (x, y) and f _y (x, y) are both continuous at (0, 0), then f (x, y) is differentiable at (0, 0).

px ² + y ² if (x, y) 6= (0, 0) 0 if (x, y) = (0, 0) .

(b) Show that f _x (0, 0) and f _y (0, 0) both exist and evaluate their values.

x ² tan ⁻¹ (y/x) − y ² tan ⁻¹ (x/y) if x 6= 0 and y 6= 0

(a) Evaluate f _x (x, y) and f _y (x, y) for x 6= 0 and y 6= 0.

(b) Evaluate f _x (0, 0) and f _y (0, 0), and show that f is differentiable at (0, 0). What is the tangent plane of the surface z = f (x, y) at (0, 0, 0)?

(c) Evaluate f _x (0, y) for y 6= 0 and f _y (x, 0) for x 6= 0. Show that f _xy (0, 0) 6= f _yx (0, 0).

D. Let f (x, y), u(x, y) and v(x, y) be C ² functions. Suppose that f _xx + f _yy = 0 and u _x = v _y , u _y = −v _x , show that the function φ(x, y) = f (u(x, y), v(x, y)) also satisfies φ xx + φ yy = 0.

A. Let u(x, y) : R ² → R be a C ² function. Express u _xx + u _yy in polar coordinate.

B. Let f (x, y) : R ² → R be a C ¹ function. Show that d dy

Hint. Every continuous function defined on a bounded and closed set D ⊂ R ⁿ is uniformly continuous.

L, where L = Adx + Bdy + Cdz defined on R ³ . Prove that L is exact, that is, L = df for some f on R ³ , if and only if the integral is independent to the path, which means that it only depends on the end points of Γ.

1 + x ² y ² over the arc of y = sin 1

(a) Show that F (x, y, z) = 0 can be solve for z = f (x, y), where f is a C ¹ function, near (0, 0, 0).

(b) Find f _x (0, 0) and f _y (0, 0). What is the tangent plane of the surface F (x, y, z) = 0 at (0, 0, 0)?

C. Find the stationary points of f (x, y) = x ³ + (y − x)(2y + x) − 3

2 y ² and determine whether they are local maximum, local minimum or saddle point.

a

. B. Calculate the first fundamental form of the surface of revolution given by r = p

x ² + y ² = f (z), where f is a C ¹ function, in terms of the cylindrical coordinates z and θ = tan ⁻¹ y

x . C. Let S be the sphere x ² + y ² + z ² = 1.

(b) Show that the parametrization r(u, v) : R ² → R ³ in (a) is conformal. That is, if two curves on z = 0, which intersect at (u, v, 0), are orthogonal at (u, v, 0), then the two image curves on the sphere are also orthogonal at r(u, v).

D. Consider the function U = F(X) = (x ² − y ² , xy).

(a) Obtain an iterative approximation G(X), which depends on given U, for the inverse transfor- mation F ⁻¹ (U) near X ₀ = (1, 1) or U ₀ = (0, 1). Verify that the fixed point X _fixed of G satisfies U = F(X _fixed ).

(b) Show that there exists a δ > 0 s.t. for any U ∈ B _δ (U ₀ ) the iteration X _n+1 = G(X _n ) with initial value X ₀ converges to a limit, denoted by X(U).

A1. Show that S is Jordan measurable in R ⁿ if and only if ∂S has Jordan measure 0 in R ⁿ .

A2. Let S and T be any disjoint sets whose union has area. Show that A ⁺ (S) + A ⁻ (T ) = A(S ∪ T ).

B. (a) Find the volume common to the two cylinders x ² + z ² ≤ 1 and y ² + z ² ≤ 1.

(b) Find the volume common to the three cylinders x ² + z ² ≤ 1, y ² + z ² ≤ 1 and x ² + y ² ≤ 1.

e ^x

(x ² + y ² + z ² )xyzdxdydz.

e ^x+y+z dxdydz.

x ⁰ = (x + y + z)/ √ 3 y ⁰ = (x − z)/ √

2 z ⁰ = (x − 2y + z)/ √

(1 + x ² + y ² + z ² ) ² converges or diverges. If it con- verges, evaluate the integral.

C. Find the volume of the n-simplex described by x k ≥ 0 for k = 1, 2, · · · , n and x ₁ a ₁ + x ₂

a ₂ + · · · + x _n a _n ≤ 1.

e ^−tx cos x dx for any fixed t.

e ^−bx − e ^−ax

1 + f _x ² + f _y ² dxdy