MA222-FinalExam-Answers

Which of the following is NOT a correct description of the sphere of radius

centered at the origin?

Choose one answer.

	a.
	b.
	c.
	d.

Question 2

Which of the following functions DOES NOT have periodic boundary conditions on

Choose one answer.

	a.
	b.
	c.
	d.

Question 3

Which of the following functions DOES NOT have periodic boundary conditions on

Choose one answer.

	a.
	b.
	c.
	d.

Question 4

What minimal combination of the following three conditions ensures the solution of the Laplace equation

on a bounded domain

with piecewise smooth boundary must be identically 0 throughout the entire domain

?
I.

II.

III.

, for all

Choose one answer.

	a. I, II, and III
	b. II only
	c. Both I and III
	d. Both II and III

Question 5

Assume

is a domain with piecewise smooth boundary,

, and

are continuous functions. Which of the following must be true of any function satisfying the nonhomogeneous Laplace equation

?
I. Either

for every

,
or

, for every

,
where are nonnegative functions and h is nontrivial.
II.

III.

Choose one answer.

	a. Both I and II
	b. III only
	c. I, II, and III
	d. I only

Question 6

Let

be a domain with piecewise smooth boundary and suppose that the continuously differentiable function

satisfies the heat equation

and is nontrivial on

. Which of the following conditions might be true?
I.

, for all

.
II.

, where

is not identically zero on

.
III.

, for all

Choose one answer.

	a. I or II
	b. I or III
	c. II or III
	d. I, II, or III

Question 7

Which of the following is an orthonormal set of vectors in

Choose one answer.

	a.
	b.
	c.
	d.

Question 8

Let

be a continuous function. Which of the following is the

norm of

Choose one answer.

	a.
	b.
	c.
	d.

Question 9

Which of the following function is NOT in

Choose one answer.

	a.
	b.
	c.
	d.

Question 10

Which of the following statements, if any, are false?
I.

, where the inner product is for the space

.
II.

is an orthonormal set of functions in

.
III.

is an orthogonal set of functions in

Choose one answer.

	a. Both I and III
	b. Both II and III
	c. III only
	d. II only

Question 11

Which of the following statements guarantees that

, where

Choose one answer.

	a. in ?.
	b. converges pointwise to .
	c. is uniformly bounded on .
	d. .

Question 12

Fill in the blank. If

and

, then

_____.

Choose one answer.

	a. -12
	b.
	c.
	d.

Question 13

Suppose

is a sequence of real-valued functions in

and that

is a given function. Which of the following statements must be true?
I.

uniformly on

.
II.

is a continuous function.
III.

Choose one answer.

	a. Both I and II
	b. Both I and III
	c. Both II and III
	d. I, II, and III

Question 14

Which of these statements, if any, is true?
I.

, where

is a positive constant, belongs to

, for any choice of

.
II. If

, then

is continuous on

Choose one answer.

	a. I only
	b. II only
	c. Both I and II
	d. Neither I nor II

Question 15

Define

Which of the following is equal to

Choose one answer.

	a. 81
	b. -2
	c. 0
	d. 1

Question 16

Which of the following statements is true?
I. If

is uniformly convergent on

, then

is convergent in the sense of

.
II. If

is convergent in the sense of

, then

is pointwise convergent on

.
III. If

is pointwise convergent on

, then

is convergent in the sense of

Choose one answer.

	a. I only
	b. I and II only
	c. I and III only
	d. I, II, and III

Question 17

Suppose that

is an orthogonal basis for

. Which of the following statements is false? (In what follows,

stands for the inner product on

and

is the standard

norm.)

Choose one answer.

	a. , for all .
	b. , for all , and for all .
	c. For all , .
	d. .

Question 18

Suppose that

is an orthonormal basis for

. Which of the following statements is true? (In what follows,

stands for the inner product on

and

is the standard

norm.)
I.

for any

.
II.

, for all

.
III.

Choose one answer.

	a. I, II, and III
	b. I and II only
	c. I and III only
	d. II and III only

Question 19

Complete the following statement. A linear operator

is self-adjoint, if and only if

is:

Choose one answer.

	a. bounded.
	b. symmetric.
	c. continuous.
	d. normal.

Question 20

is the Fourier sine series of

, which of the following statements must be true?
I. The graphs of

better approximate the function

at points of continuity in

gets larger.
II.

, for any

.
III.

, for any

Choose one answer.

	a. I, II, and III
	b. I and II only
	c. II and III only
	d. I only

Question 21

Which of the following statements is false?

Choose one answer.

	a. Every function f in has a unique Fourier representation on .
	b. There is a piecewise continuous function defined on that does not have a unique Fourier representation.
	c. , where the norm is taken in the sense of .
	d. If , then the Fourier sine series representation of converges pointwise to .

Question 22

is the Fourier cosine series representation for

, then

_________.

Choose one answer.

	a.
	b.
	c.
	d.

Question 23

Which of the following is true concerning the vector field

defined by

Choose one answer.

	a. , for some .
	b. is orthogonal to the vector , for all .
	c. has a positive divergence at every point .
	d. is a conservative vector field.

Question 24

Suppose that

Which of the following is a correct characterization of the Fourier cosine series of

Choose one answer.

	a. It is not defined, because of the discontinuity of at .
	b. It converges to , because .
	c. It converges to .
	d. It does not converge because of the jump discontinuity.

Question 25

is the Fourier sine series for

, which of the following is true?
I.

II.

Choose one answer.

	a. Neither I nor II
	b. I only
	c. II only
	d. Both I and II

Question 26

Which of the following statements is true?
I. The Fourier sine series of

is simply

.
II. The Fourier cosine series of

is simply

Choose one answer.

	a. I only
	b. II only
	c. Neither I nor II
	d. Both I and II

Question 27

Which of these formulas is equal to

, where

Choose one answer.

	a.
	b.
	c.
	d.

Question 28

Which of the following is an accurately-stated consequence of Gibb’s phenomenon for a function

Choose one answer.

	a. The Fourier cosine series of on does not converge to at .
	b. The Fourier sine series of is undefined at points of jump discontinuity of in .
	c. The Fourier sine series of does not converge to when is continuous at but has a sharp corner there.
	d. The Fourier cosine series of does not converge uniformly to on .

Question 29

Suppose the real Fourier series for

is given by

.
Which of the following statements is false?
I. If

, then this series converges pointwise on

.
II.

III. If

, for any

, then

is an even function.

Choose one answer.

	a. I only
	b. II only
	c. I and III only
	d. II and III only

Question 30

Suppose

is the complex Fourier series representation of

, and assume that

. Which of the following is false?

Choose one answer.

	a. , for any
	b. for any
	c.
	d.

Question 31

Assume that the Fourier sine series representation of

. Which of the following is the unique solution of the one dimensional heat equation

coupled with homogenous Dirichlet boundary conditions

and initial condition

Choose one answer.

	a.
	b.
	c.
	d.

Question 32

Assume that the Fourier sine series representation of f is

. Which of the following is the unique solution of the one dimensional heat equation

coupled with homogenous Neumann boundary conditions

and initial condition

Choose one answer.

	a.
	b.
	c.
	d.

Question 33

Let

be defined by

. Which of the following is the Fourier transform

Choose one answer.

	a.
	b.
	c.
	d.

Question 34

Which of the following functions is harmonic in

Choose one answer.

	a.
	b.
	c.
	d.

Question 35

Suppose f and g are continuous and absolutely integrable on

. Which of the following statements is true?
I.

II.

if and only if

Choose one answer.

	a. I only
	b. II only
	c. Neither I nor II
	d. Both I and II

Question 36

Suppose

and its Fourier transform

also belongs to

. Which of the following need NOT hold?

Choose one answer.

	a. is continuous on .
	b. is differentiable on .
	c. , for every .
	d. .

Question 37

Suppose

and denote its Fourier transform by

. Which of the following statements are true?
I.

is continuous on

.
II.

is bounded on

.
III.

Choose one answer.

	a. I only
	b. I and III only
	c. II and III only
	d. I, II, and III

Question 38

Suppose

and

, and denote their Fourier transforms by

and

, respectively. Which of the following are accurate statements concerning the convolution

?
I. If

then

.
II.

Choose one answer.

	a. I only
	b. II only
	c. Neither I nor II
	d. Both I and II

Question 39

Fill in the blank. Suppose that

and define a function

. Denote their Fourier transforms by

and

, respectively. For any

_______.

Choose one answer.

	a.
	b.
	c.
	d.

Question 40

Fill in the blank. Suppose that

and define a function

. Denote their Fourier transforms by

and

, respectively. Then,

_________.

Choose one answer.

	a.
	b.
	c.
	d.

Question 41

Fill in the blank. Suppose that

and define a function

. Denote their Fourier transforms by

and

, respectively. Then,

__________.

Choose one answer.

	a.
	b.
	c.
	d.

Question 42

Let

be a function. Which of the following descriptive statements regarding its Fourier transform

are correct?
I. The Fourier transform converts differentiation in

into multiplication in

.
II. The smoothness of

is closely related to the asymptotic decay rate of its Fourier transform

?
III. The asymptotic decay rate of

determines the smoothness of its Fourier transform.

Choose one answer.

	a. I only
	b. I and II only
	c. I and III only
	d. I, II, and III

Question 43

Which of the following is the Fourier transform of

Choose one answer.

	a.
	b.
	c.
	d.

Question 44

Which of the following is NOT equal to the Gauss-Weierstrass kernel

Choose one answer.

	a.
	b. , where is the Fourier transform of .
	c.
	d.

Question 45

Define the operator

.
Which of the following is NOT a characteristic of the operator

Choose one answer.

	a.
	b. , for all . (Here, T stands for transpose.)
	c. does not contain the origin . (Here, T stands for transpose.)
	d. is a linear operator.

Question 46

Let

and

be the Gauss-Weierstrass kernel. Consider the one dimensional heat equation

coupled with the initial condition

. Which of the following is the unique solution of this IVP?

Choose one answer.

	a.
	b.
	c.
	d.

Question 47

Suppose that

and denote their Fourier transforms by

and

, respectively. Which of the following is the unique solution of the one dimensional wave equation

coupled with the initial conditions

Choose one answer.

	a.
	b.
	c.
	d.

Question 48

Suppose that

is a function of three variables,

, and

. Which of the following is the spherical average of

of radius

? (Here,

Choose one answer.

	a.
	b.
	c.
	d.

Question 49

Let

denote the spherical average of

of radius

and let

. Which of the following is the unique solution of the three-dimensional wave equation

coupled with the initial conditions

Choose one answer.

	a.
	b.
	c.
	d.

Question 50

Assume that

with Fourier transform

. Which of the following is the unique solution

of the two-dimensional Laplace equation

, where

coupled with the boundary condition

Choose one answer.

	a.
	b.
	c.
	d.

Question 51

Let

with Fourier cosine transform

and Fourier sine transform

. Which of the following is the unique solution of the one dimensional heat equation

coupled with the initial condition

and boundary condition

Choose one answer.

	a.
	b.
	c.
	d.

Question 52

Suppose

is a domain and

is a nonconstant harmonic function. Which of these is true?
I.

.
II.

has no maximal points on the boundary of

Choose one answer.

	a. I only
	b. II only
	c. Neither I nor II
	d. Both I and II

Question 53

Which of the following is the d’Alembert solution of the one-dimensional wave equation

coupled with the initial conditions

and

Choose one answer.

	a.
	b.
	c.
	d.

Question 54

Let

denote the Fourier transform of

. Which of the following is false?

Choose one answer.

	a.
	b. The inverse Fourier transform of is .
	c.
	d.

Question 55

Let

denote the Fourier transform of

. Which of the following is false?
I.

II.

III.

Choose one answer.

	a. I only
	b. III only
	c. I and II only
	d. II and III only

Question 56

Which of the following statements is true?

Choose one answer.

	a. The sum of two solutions of the PDE is also a solution.
	b. The PDE is homogeneous.
	c. The 3-dimensional Laplace equation is a nonlinear elliptic PDE.
	d. The Fokker-Plank equation is nonhomogeneous, where .

Question 57

Let

and

denote the Fourier sine and cosine transforms of

, respectively. Which of the following holds?
I.

II.

Choose one answer.

	a. I only
	b. II only
	c. Both I and II
	d. Neither I nor II

Question 58

Which of the following PDEs are NOT evolution equations?
I.

II.

III.

Choose one answer.

	a. Both II and III
	b. Both I and III
	c. I, II, and III
	d. I only

Question 59

Assume

represents the temperature along a thin wire at position x at time t, where

and

. Which of the following statements is false when viewed as a condition coupled with the heat equation

Choose one answer.

	a. , where stands for the outward normal derivative, is a Neumann boundary condition.
	b. are homogeneous Dirichlet boundary conditions.
	c. are initial conditions.
	d. is a periodic initial condition.

Question 60

Which of the following statements is an accurate physical interpretation of a homogeneous Neumann boundary condition in the context of the heat equation?

Choose one answer.

	a. The boundary is an imperfect conductor.
	b. The boundary is a permeable barrier to the flow of heat.
	c. No heat is crossing the boundary.
	d. The speed and direction of heat flow through the left boundary are positive.