Calculus of One Variable

Summary

This exam will address fundamental topics from single-variable calculus, including the concepts of conti-nuity and differentiability and the explicit calculation of limits, derivatives, and integrals.

Sample Questions

Here’s an example of a conceptual question that appeared on a prior exam:

Consider the function

g(x) =

{x2 if x is rational

0 if x is irrational.

Is g(x) continuous at x = 0? Is g(x) differentiable at x = 0? Justify your responses.

In order to answer this question correctly, you need to understand the notions of continuity and differentia-bility rigorously, not merely intuitively. Note that explicit justification is required to support each correctresponse.

Here are examples of computational questions that have appeared on prior exams:

Compute limx→e

(1−ln xx−e

)2

.

Compute the arclength of the graph of f(x) = coshx between x = 0 an x = 1.

In order to earn full credit, your response to either question would have to be justified by an explicit, step-by-step calculation.

References

Single-variable calculus is covered in many undergraduate textbooks. It’s a good idea to review more thanone book in preparation for this exam.

Calculus of Several Variables

Summary

This exam will address topics from vector calculus in two and three dimensions, including gradients ofscalar functions, computations involving the ∇ operator, line, surface, and volume integrals, and variationsof Stokes’ theorem (including Green’s theorem and the divergence theorem).

Sample Questions

Here’s a prior year’s exam in its entirety:

Let (r, θ) be polar coordinates on the plane z = 0. Let Ω denote the region bounded below bythis plane and above by the surface Σ on which z = (1− r + cos θ)er, as shown:

Let C denote the curve along which the plane z = 0 intersects Σ, and let

F = (y + z cosx)ı+ (x+ z sin y)j+ (sinx− cos y)k

andG = x(y + z)ı+ yzj− (yz + z2)k

be two vector fields.

1. Compute the length of C.

2. Compute the contour integral∫CF · ds in the direction of increasing θ.

3. Compute the net flux into Ω of the vector field G.

4. Is F conservative? Is G? Justify your responses.

Note that variations of Stokes’ theorem can be used to simplify the second and third questions substantially.

References

Vector calculus is covered in many textbooks, including Erwin Kreyszig’s Advanced Engineering Mathe-matics. It’s a good idea to review more than one book in preparation for this exam.

ORDINARY DIFFERENTIAL EQUATIONS

1. Find the general solution to

2. Find the general solution to the system of equations:

PARTIAL DIFFERENTIAL EQUATIONS

1. Solve the initial value problem

- < x <

- < x <

using Fourier transforms.

2. Suppose u1 and u2 are solutions of the following equations. For which equations is u1 + u2 a solution?

a) b) c) d) e)

Ph.D. Qualifying Exam Mathematics – Numerical Methods Oct. 2010

1. Given two vectors V1 = 72i – 120j – 8k and V2 = – i – 3j + 4k(a) Find the angle between V1 and V2.(b) Find the unit vector n that is perpendicular to both V1 and V2.(c) Represent V1 using two vector components, V1x and V1y, one parallel and the other

perpendicular to V2.

2.(a) Calculate the scalar triple product of three vectors v1 = 2i + 6j – 7k, v2 = –i + 7j – 4k, and

v3 = 5i – 6j + 4k. Determine if the three vectors are linearly independent.(b) Determine the function of line L1 that passes though point (1, –3, –2) with direction

vector d1 = 2i + 5j + 3k(c) For a surface given by 3x2 + 4xy – 7z3 = 2, the surface normal is given by

n = f = <6x+4y, 4x, –21z2>.Obtain the tangent plane of the surface at point (1, –2, –1).

3. Evaluate 1

0

2 43 dxxxI by

(a) the trapezoidal rule using two intervals.(b) the Simpson’s 1/3 rule using two intervals. What is the error reduction if the interval size

is reduced by half?(c) What is the order of error reduction in the trapezoidal rule if the interval size is reduced

by half? What is the order of error reduction in the Simpson’s 1/3 rule if the interval size is reduced by half?

4. For the following system of equations with four unknowns x1, x2, x3, and x43x1 – 2x2 + x3 + 2x4 = 17x1 + 3x2 + 7x3 – x4 = –18–2x1 + x2 + x3 + 3x4 = 34x1 + 2x2 + 3x3 – x4 = 0

(a) Represent the above equations in the standard matrix form [A]{x} = {b} and use Gaussian elimination to obtain the coefficient matrix [A] in the upper triangular form. Show your work.

(b) Use back-substitution to solve for {x}. Show your work.

with some vector algebra, too

Q1) Given two arbitrary square (n x n), nonsingular matrices A and B, prove det(sI-AB) = det(sI – BA), where I is an n x n identity matrix and s is any scalar. [Try and use this hint: Two n x n matrices M and N are similar if M = P-1NP for some invertible n x n matrix P]. ‘det’ stands for determinant. (35) Q2) Let S be a symmetric matrix and A be a skew-symmetric matrix. Prove that trace(SA) = trace(AS) = 0 [Trace of an n x n matrix is defined as sum of all elements on its main diagonal (i.e. the diagonal from the upper left to the lower right)] (35) Q3) Find the surface in terms of a,b,c on which the following system of equations has no solution

Could there be any values of a,b,c,d for which the system has infinite solution? (Justify).

(30)

Linear Algebra

Section: Vector Algebra

Description: Algebraic and geometric properties of vectors in three dimensions, dot and cross products, linear independence of vectors, scalar triple product, equations of lines and planes.

Question 1: Use the following form for the equation of a line: tbbb

aaa

zyx

3

2

1

3

2

1

Find an equation for the line that contains the points 5,6,3 and 7,1,9

Write the equations for the a and b parameters that are satisfied if and only if two lines are equivalent.

(for line 1, use 321321 ,,,,, bbbaaa and for line 2, use 321321 ,,,,, bbbaaa )

Question 2: 130

,132

,321

321 vvv

Show whether the following statement is true or false by performing the calculations: 3121321 vvvvvvv

Why would you expect the above to be true or false?

Find the projection of v1 onto v2.

Find the angle between v2 and v3.

Find a value of s so that the vector 1

24

sss

v is perpendicular to v2.

Ph.D. Math Qualifying Exam - Complex Variables - Spring 2010

Please answer both questions

1) In 2-D potential flow theory, the complex potential, F = F (z), is related topolar-cylindrical velocity components via the following formula:

dF

dz= vr(r, θ) − ivθ(r, θ) (1)

where vr and vθ are the r and θ velocity components, respectively, and where

z = reiθ

Note, that for an analytic potential function, F,

dF

dz=

∂F

∂r=

1ir

∂F

∂θ

a) Given

F (z) =−iΓ2π

ln reiθ

find the corresponding r− and θ− velocity components.

b) the complex potential is also related to the so-called velocity potential, φ, andthe stream function, ψ, according to

F (z) = φ(r, θ) + iψ(r, θ) (2)

Streamlines in a flow, in turn, correspond to lines on which ψ is constant, i.e., oneach streamline ψ assumes a constant value. Sketch a few of the streamlines inthe above flow.

1

2) Consider an oscillator, e.g., a spring-mass system, driven by a periodic forcingfunction:

x + a2x = b cos ωt (3)

where x = x(t) is the instantaneous position of the mass, a2 and b are constants,and ω is the driving frequency.

a) In order to determine the (cyclically) steady motion, x(t), first (quickly) show/confirmthat

cos ωt =12(eiωt + e−iωt) (4)

b) Next, assume a solution of the form x(t) = x1(t) + x2(t), where x1(t) = Aeiωt

and x2(t) = Be−iωt. Solve for x1(t) first by plugging the assumed form of x1(t)into Eq. (3) and by replacing the right side of (3) with the product of b and thefirst term on the right side of Eq. (4). Do the same to obtain x2(t), replacing theright side of (3) with the product of b and the second term on the right of (4).

c) Quickly confirm that the assumed solution, x = x1 + x2, satisfies the governingequation, Eq. (3).

2

1

1. a) Explain the circumstances for applying a two-sample, one and two-sided ‘t’ test

b) Table 1 shows the measured mean viscosities of two different brands of car oil.

Test the hypothesis that the two brands are equal at a 10% significance level,

Table 1: Measured value of the mean viscosities of two brands of car oil

Brand A 1062 1058 1033 1072 1044 1074

Brand B 1050 1052 1058 1062 1055 1051 1053

2. State the integral equation for computing the mean and variance based on a knowledge

of the probability density function p x .

For a triangular probability density function of width 2a determine the standard

deviation

probability and statistics

2

3

4

5

Where appropriate, the following equations may be used

The probability mass function for a binomial random variable (n, p) is

!!!

1)(

iinn

in

ppin

iXP ini

Mean xN

f xi i1 where N fi

Variance2 2 2 21 1

Nf x x

Nf x xi i i i

Variance of the mean computed from a normally distributed sample

xx

N2

2

Standard normalized variate u[0,1]

u x x

Poisson distribution

er

rxPr

!)(

mean of x = variance =

Chi-squared2

2

2

1( )n s

22

1

O EE

i i

ii

n

‘t’ test

t xs n

zx x

sn nc

1 2

1 2

1 1

211

21

222

2112

nnsnsnsc

Fischer’s ‘F’ test

F

s

sn

n1 2

12

12

22

22

12

1

22

2

12

1

22

2

1

1,