BSc: Mathematical Analysis II

MathematicalAnalysis II

• Course name: Mathematical Analysis II
• Course number: BS-01

Course Characteristics

Key concepts of the class

• Multivariate calculus: derivatives, differentials, maxima and minima
• Multivariate integration
• Functional series. Fourier series
• Integrals with parameters

What is the purpose of this course?

The goal of the course is to study basic mathematical concepts that will be required in further studies. The course is based on Mathematical Analysis I, and the concepts studied there are widely used in this course. The course covers differentiation and integration of functions of several variables. Some more advanced concepts, as uniform convergence of series and integrals, are also considered, since they are important for understanding applicability of many theorems of mathematical analysis. In the end of the course some useful applications are covered, such as gamma-function, beta-function, and Fourier transform.

- What should a student remember at the end of the course?

By the end of the course, the students should be able to:

• find partial and directional derivatives of functions of several variables;
• find maxima and minima for a function of several variables
• use Fubini’s theorem for calculating multiple integrals
• calculate line and path integrals
• distinguish between point wise and uniform convergence of series and improper integrals
• decompose a function into Fourier series
• calculate Fourier transform of a function

- What should a student be able to understand at the end of the course?

By the end of the course, the students should be able to understand:

• how to find minima and maxima of a function subject to a constraint
• how to represent double integrals as iterated integrals and vice versa
• what the length of a curve and the area of a surface is
• properties of uniformly convergent series and improper integrals
• beta-function, gamma-function and their properties
• how to find Fourier transform of a function

- What should a student be able to apply at the end of the course?

By the end of the course, the students should be able to ...

• find multiple, path, surface integrals
• find the range of a function in a given domain
• decompose a function into Fourier series

Course evaluation

Course grade breakdown

		Proposed points
Test 1	?	10
Midterm	?	25
Test 2	?	10
Participation	?	5
Final exam	?	50

Grades range

Course grading range

		Proposed range
A. Excellent	90-100	85-100
B. Good	75-89	65-84
C. Satisfactory	60-74	45-64
D. Poor	0-59	0-44

Resources and reference material

• Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson
• Jerrold Marsden, Alan Weinstein (1985) Calculus (in three volumes; volumes 2 and 3), Springer

Course Sections

The main sections of the course and approximate hour distribution between them is as follows:

Course Sections

Section	Section Title	Teaching Hours
1	Differential Analysis of Functions of Several Variables	24
2	Integration of Functions of Several Variables	30
3	Uniform Convergence of Functional Series. Fourier Series	18
4	Integrals with Parameter(s)	18
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Section 1

Section title:

Differential Analysis of Functions of Several Variables

Topics covered in this section:

• Limits of functions of several variables
• Partial and directional derivatives of functions of several variables. Gradient
• Differentials of functions of several variables. Taylor formula
• Maxima and minima for functions of several variables
• Maxima and minima for functions of several variables subject to a constraint

What forms of evaluation were used to test students’ performance in this section?

	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Find ${\textstyle \lim \limits _{x\to 0}\lim \limits _{y\to 0}u(x;y)}$, ${\textstyle \lim \limits _{y\to 0}\lim \limits _{x\to 0}u(x;y)}$ and ${\textstyle \lim \limits _{(x;y)\to (0;0)}u(x;y)}$ if ${\textstyle u(x;y)={\frac {x^{2}y+xy^{2}}{x^{2}-xy+y^{2}}}}$.
2. Find the differential of a function: (a) ${\textstyle u(x;y)=\ln \left(x+{\sqrt {x^{2}+y^{2}}}\right)}$; (b) ${\textstyle u(x;y)=\ln \sin {\frac {x+1}{\sqrt {y}}}}$.
3. Find the differential of ${\textstyle u(x;y)}$ given implicitly by an equation ${\textstyle x^{3}+2y^{3}+u^{3}-3xyu+2y-3=0}$ at points ${\textstyle M(1;1;1)}$ and ${\textstyle N(1;1;-2)}$.
4. Find maxima and minima of a function subject to a constraint (or several constraints):
   1. ${\textstyle u=x^{2}y^{3}z^{4}}$, ${\textstyle 2x+3y+4z=18}$, ${\textstyle x>0}$, ${\textstyle y>0}$, ${\textstyle z>0}$;
   2. ${\textstyle u=x-y+2z}$, ${\textstyle x^{2}+y^{2}+2z^{2}=16}$;
   3. ${\textstyle u=\sum \limits _{i=1}^{k}a_{i}x_{i}^{2}}$, ${\textstyle \sum \limits _{i=1}^{k}x_{i}=1}$, ${\textstyle a_{i}>0}$;

Typical questions for seminar classes (labs) within this section

1. Let us consider ${\textstyle u(x;y)={\begin{cases}1,&x=y^{2},\\0,&x\neq y^{2}.\end{cases}}}$ Show that this function has a limit at the origin along any straight line that passes through it (and all these limits are equal to each other), yet this function does not have limit as ${\textstyle (x;y)\to (0;0)}$.
2. Find the largest possible value of directional derivative at point ${\textstyle M(1;-2;-3)}$ of function ${\textstyle f=\ln xyz}$.
3. Find maxima and minima of functions ${\textstyle u(x,y)}$ given implicitly by the equations:
   1. ${\textstyle x^{2}+y^{2}+u^{2}-4x-6y-4u+8=0}$, ${\textstyle u>2}$;
   2. ${\textstyle x^{3}-y^{2}+u^{2}-3x+4y+u-8=0}$.
4. Find maxima and minima of functions subject to constraints:
   1. ${\textstyle u=xy^{2}}$, ${\textstyle x+2y-1=0}$;
   2. ${\textstyle u=xy+yz}$, ${\textstyle x^{2}+y^{2}=2}$, ${\textstyle y+z=2}$, ${\textstyle y>0}$.

Test questions for final assessment in this section

1. Find all points where the differential of a function ${\textstyle f(x;y)=(5x+7y-25)e^{-x^{2}-xy-y^{2}}}$ is equal to zero.
2. Show that function ${\textstyle \varphi =f\left({\frac {x}{y}};x^{2}+y-z^{2}\right)}$ satisfies the equation ${\textstyle 2xz\varphi _{x}+2yz\varphi _{y}+\left(2x^{2}+y\right)\varphi _{z}=0}$.
3. Find maxima and minima of function ${\textstyle u=2x^{2}+12xy+y^{2}}$ under condition that ${\textstyle x^{2}+4y^{2}=25}$. Find the maximum and minimum value of a function
4. ${\textstyle u=\left(y^{2}-x^{2}\right)e^{1-x^{2}+y^{2}}}$ on a domain given by inequality ${\textstyle x^{2}+y^{2}\leq 4}$;

Section 2

Section title:

Integration of Functions of Several Variables

Topics covered in this section:

• Z-test
• Double integrals. Fubini’s theorem and iterated integrals
• Substituting variables in double integrals. Polar coordinates
• Triple integrals. Use of Fubini’s theorem
• Spherical and cylindrical coordinates
• Path integrals
• Area of a surface
• Surface integrals

What forms of evaluation were used to test students’ performance in this section?

	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: ${\textstyle \iint \limits _{D}f(x;y)\,dx\,dy}$ where ${\textstyle D=\left\{(x;y)\left|x^{2}+y^{2}\leq 9,\,x^{2}+(y+4)^{2}\geq 25\right.\right\}}$.
2. Represent integral ${\textstyle I=\displaystyle \iiint \limits _{D}f(x;y;z)\,dx\,dy\,dz}$ as iterated integrals with all possible (i.e. 6) orders of integration; ${\textstyle D}$ is bounded by ${\textstyle x=0}$, ${\textstyle x=a}$, ${\textstyle y=0}$, ${\textstyle y={\sqrt {ax}}}$, ${\textstyle z=0}$, ${\textstyle z=x+y}$.
3. Find line integrals of a scalar fields ${\textstyle \displaystyle \int \limits _{\Gamma }(x+y)\,ds}$ where ${\textstyle \Gamma }$ is boundary of a triangle with vertices ${\textstyle (0;0)}$, ${\textstyle (1;0)}$ and ${\textstyle (0;1)}$.

Typical questions for seminar classes (labs) within this section

1. Change order of integration in the iterated integral ${\textstyle \int \limits _{0}^{\sqrt {2}}dy\int \limits _{y}^{\sqrt {4-y^{2}}}f(x;y)\,dx}$.
2. Find the volume of a solid given by ${\textstyle 0\leq z\leq x^{2}}$, ${\textstyle x+y\leq 5}$, ${\textstyle x-2y\geq 2}$, ${\textstyle y\geq 0}$.
3. Change into polar coordinates and rewrite the integral as a single integral: ${\textstyle \displaystyle \iint \limits _{G}f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx\,dy}$, ${\textstyle G=\left\{(x;y)\left|x^{2}+y^{2}\leq x;\,x^{2}+y^{2}\leq y\right.\right\}}$.
4. Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at ${\textstyle A}$ and finishes at ${\textstyle B}$: ${\textstyle \displaystyle \int \limits _{\Gamma }\left(x^{4}+4xy^{3}\right)\,dx+\left(6x^{2}y^{2}-5y^{4}\right)\,dy}$, ${\textstyle A(-2;-1)}$, ${\textstyle B(0;3)}$;

Test questions for final assessment in this section

1. Domain ${\textstyle G}$ is bounded by lines ${\textstyle y=2x}$, ${\textstyle y=x}$ and ${\textstyle y=2}$. Rewrite integral ${\textstyle \iint \limits _{G}f(x)\,dx\,dy}$ as a single integral.
2. Represent the integral ${\textstyle \displaystyle \iint \limits _{G}f(x;y)\,dx\,dy}$ as iterated integrals with different order of integration in polar coordinates if ${\textstyle G=\left\{(x;y)\left|a^{2}\leq x^{2}+y^{2}\leq 4a^{2};\,|x|-y\geq 0\right.\right\}}$.
3. Find the integral making an appropriate substitution: ${\textstyle \displaystyle \iiint \limits _{G}\left(x^{2}-y^{2}\right)\left(z+x^{2}-y^{2}\right)\,dx\,dy\,dz}$, ${\textstyle G=\left\{(x;y;z)\left|x-1<y<x;\,1-x<y<2-x;\,1-x^{2}+y^{2}<z<y^{2}-x^{2}+2x\right.\right\}}$.
4. Use divergence theorem to find the following integrals ${\textstyle \displaystyle \iint \limits _{S}(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy}$ where ${\textstyle S}$ is the outer surface of a tetrahedron ${\textstyle {\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}\leq 1}$, ${\textstyle x\geq 0}$, ${\textstyle y\geq 0}$, ${\textstyle z\geq 0}$;

Section 3

Section title:

Uniform Convergence of Functional Series. Fourier Series

Topics covered in this section:

• Uniform and point wise convergence of functional series
• Properties of uniformly convergent series
• Fourier series. Sufficient conditions of convergence and uniform convergence
• Bessel’s inequality and Parseval’s identity.

What forms of evaluation were used to test students’ performance in this section?

	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. ${\textstyle \sum \limits _{n=1}^{\infty }e^{-n\left(x^{2}+2\sin x\right)}}$, ${\textstyle \Delta _{1}=(0;1]}$, ${\textstyle \Delta _{2}=[1;+\infty )}$;
2. ${\textstyle \sum \limits _{n=1}^{\infty }{\frac {\sqrt {nx^{3}}}{x^{2}+n^{2}}}}$, ${\textstyle \Delta _{1}=(0;1)}$, ${\textstyle \Delta _{2}=(1;+\infty )}$
3. Show that sequence ${\textstyle f_{n}(x)=nx\left(1-x\right)^{n}}$ converges non-uniformly on ${\textstyle [0;1]}$ to a continuous function ${\textstyle f(x)}$, but ${\textstyle \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x)\,dx=\lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x)\,dx}$.
4. Decompose the following function determined on ${\textstyle [-\pi ;\pi ]}$ into Fourier series using the standard trigonometric system ${\textstyle \left.\left\{1;\cos kx;\sin kx\right\}\right|_{k=1}^{\infty }}$. Draw the graph of the sum of Fourier series obtained. ${\textstyle f(x)={\begin{cases}1,\;0\leq x\leq \pi ,\\0,\;-\pi \leq x<0.\end{cases}}}$
5. Prove that if for an absolutely integrable function ${\textstyle f(x)}$ on ${\textstyle [-\pi ;\pi ]}$
   1. ${\textstyle f(x+\pi )=f(x)}$ then ${\textstyle a_{2k-1}=b_{2k-1}=0}$, ${\textstyle k\in \mathbb {N} }$;
   2. ${\textstyle f(x+\pi )=-f(x)}$ then ${\textstyle a_{0}=0}$, ${\textstyle a_{2k}=b_{2k}=0}$, ${\textstyle k\in \mathbb {N} }$.

Typical questions for seminar classes (labs) within this section

1. Show that sequence ${\textstyle f_{n}(x)=nx\left(1-x^{2}\right)^{n}}$ converges on ${\textstyle [0;1]}$ to a continuous function ${\textstyle f(x)}$, and at that ${\textstyle \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x)\,dx\neq \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x)\,dx}$.
2. Show that sequence ${\textstyle f_{n}(x)=x^{3}+{\frac {1}{n}}\sin \left(nx+{\frac {n\pi }{2}}\right)}$ converges uniformly on ${\textstyle \mathbb {R} }$, but ${\textstyle \left(\lim \limits _{n\rightarrow +\infty }f_{n}(x)\right)'\neq \lim \limits _{n\rightarrow +\infty }f'_{n}(x)}$.
3. Decompose ${\textstyle \cos \alpha x}$, ${\textstyle \alpha \notin \mathbb {Z} }$ into Fourier series on ${\textstyle [-\pi ;\pi ]}$. Using this decomposition prove that ${\textstyle \cot y={\frac {1}{y}}+\sum \limits _{k=1}^{\infty }{\frac {2y}{y^{2}-\pi ^{2}k^{2}}}}$.
4. Function ${\textstyle f(x)}$ is absolutely integrable on ${\textstyle [0;\pi ]}$, and ${\textstyle f(\pi -x)=f(x)}$. Prove that
   1. if it is decomposed into Fourier series of sines then ${\textstyle b_{2k}=0}$, ${\textstyle k\in \mathbb {N} }$;
   2. if it is decomposed into Fourier series of cosines then ${\textstyle a_{2k-1}=0}$, ${\textstyle k\in \mathbb {N} }$.
5. ## Decompose ${\textstyle f(x)={\begin{cases}1,\;|x|<\alpha ,\\0,\;\alpha \leqslant |x|<\pi \end{cases}}}$ into Fourier series using the standard trigonometric system.
   1. Using Parseval’s identity find ${\textstyle \sigma _{1}=\sum \limits _{k=1}^{\infty }{\frac {\sin ^{2}k\alpha }{k^{2}}}}$ and ${\textstyle \sigma _{2}=\sum \limits _{k=1}^{\infty }{\frac {\cos ^{2}k\alpha }{k^{2}}}}$.

Test questions for final assessment in the course

1. Find out whether the following functional series converge uniformly on the indicated intervals. Justify your answer. ${\textstyle \sum \limits _{n=1}^{\infty }{\frac {xn+{\sqrt {n}}}{n+x}}\ln \left(1+{\frac {x}{n{\sqrt {n}}}}\right)}$, ${\textstyle \Delta _{1}=(0;1)}$, ${\textstyle \Delta _{2}=(1;+\infty )}$;
2. Show that sequence ${\textstyle f_{n}(x)={\frac {\sin nx}{\sqrt {n}}}}$ converges uniformly on ${\textstyle \mathbb {R} }$ to a differentiable function ${\textstyle f(x)}$, and at that ${\textstyle \lim \limits _{n\rightarrow +\infty }f'_{n}(0)\neq f'(0)}$.

Section 1

Section title:

Integrals with Parameter(s)

Topics covered in this section:

• Definite integrals with parameters
• Improper integrals with parameters. Uniform convergence
• Properties of uniformly convergent integrals
• Beta-function and gamma-function
• Fourier transform

What forms of evaluation were used to test students’ performance in this section?

	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Find out if ${\textstyle \displaystyle \int \limits _{0}^{1}\left(\lim \limits _{\alpha \to 0}{\frac {2x\alpha ^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}\right)\,dx=\lim \limits _{\alpha \to 0}\int \limits _{0}^{1}{\frac {2x\alpha ^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}\,dx}$.
2. Differentiating the integrals with respect to parameter ${\textstyle \varphi }$, find it: ${\textstyle I(\alpha )=\int \limits _{0}^{\pi /2}\ln \left(\alpha ^{2}-\sin ^{2}\varphi \right)\,d\varphi }$, ${\textstyle \alpha >1}$.
3. Prove that the following integral converges uniformly on the indicated set. ${\textstyle \displaystyle \int \limits _{0}^{+\infty }e^{-\alpha x}\cos 2x\,dx}$, ${\textstyle \Delta =[1;+\infty )}$;
4. It is known that Dirichlet’s integral ${\textstyle \int \limits _{0}^{+\infty }{\frac {\sin x}{x}}\,dx}$ is equal to ${\textstyle {\frac {\pi }{2}}}$. Find the values of the following integrals using Dirichlet’s integral
   1. ${\textstyle \int \limits _{0}^{+\infty }{\frac {\sin }{\alpha x}}x\,dx}$, ${\textstyle \alpha \neq 0}$;
   2. ${\textstyle \int \limits _{0}^{+\infty }{\frac {\sin x-x\cos x}{x^{3}}}\,dx}$.

Typical questions for seminar classes (labs) within this section

1. Find out if ${\textstyle \displaystyle \int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha )\,d\alpha \right)\,dx=\int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha )\,dx\right)\,d\alpha }$ if ${\textstyle f(x;\alpha )={\frac {\alpha -x}{(\alpha +x)^{3}}}}$.
2. Find ${\textstyle \Phi '(\alpha )}$ if ${\textstyle \Phi (\alpha )=\int \limits _{1}^{2}{\frac {e^{\alpha x^{2}}}{x}}\,dx}$.
3. Differentiating the integral with respect to parameter ${\textstyle \alpha }$, find it: ${\textstyle I(\alpha )=\int \limits _{0}^{\pi }{\frac {1}{\cos x}}\ln {\frac {1+\alpha \cos x}{1-\alpha \cos x}}\,dx}$, ${\textstyle |\alpha |<1}$.
4. Find Fourier transform of the following functions:
   1. ${\textstyle f(x)={\begin{cases}1,&|x|\leq 1,\\0,&|x|>1;\end{cases}}}$
5. Let ${\textstyle {\widehat {f}}(y)}$ be Fourier transform of ${\textstyle f(x)}$. Prove that Fourier transform of ${\textstyle e^{i\alpha x}f(x)}$ is equal to ${\textstyle {\widehat {f}}(y-\alpha )}$, ${\textstyle \alpha \in \mathbb {R} }$.

Test questions for final assessment in this section

1. Find out if ${\textstyle \displaystyle \int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha )\,d\alpha \right)\,dx=\int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha )\,dx\right)\,d\alpha }$ if ${\textstyle f(x;\alpha )={\frac {\alpha ^{2}-x^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}}$.
2. Find ${\textstyle \Phi '(\alpha )}$ if ${\textstyle \Phi (\alpha )=\int \limits _{0}^{\alpha }{\frac {\ln(1+\alpha x)}{x}}\,dx}$.
3. Prove that the following integral converges uniformly on the indicated set. ${\textstyle \displaystyle \int \limits _{-\infty }^{+\infty }{\frac {\cos \alpha x}{4+x^{2}}}\,dx}$, ${\textstyle \Delta =\mathbb {R} }$;
4. Find Fourier integral for ${\textstyle f(x)={\begin{cases}1,&|x|\leq \tau ,\\0,&|x|>\tau ;\end{cases}}}$