BSc: Mathematical Analysis II

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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.

Course Objectives Based on Bloom’s Taxonomy

- 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
hline

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim\limits_{x\to0}\lim\limits_{y\to0}u(x;y)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim\limits_{y\to0}\lim\limits_{x\to0}u(x;y)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim\limits_{(x;y)\to(0;0)}u(x;y)} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u(x;y)=\frac{x^2y+xy^2}{x^2-xy+y^2}} .
  2. Find the differential of a function: (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u(x;y)=\ln\left(x+\sqrt{x^2+y^2}\right)} ; (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u(x;y)=\ln\sin\frac{x+1}{\sqrt y}} .
  3. Find the differential of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u(x;y)} given implicitly by an equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x^3+2y^3+u^3-3xyu+2y-3=0} at points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle M(1;1;1)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle N(1;1;-2)} .
  4. Find maxima and minima of a function subject to a constraint (or several constraints):
    1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u=x^2y^3z^4} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle 2x+3y+4z=18} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x>0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y>0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z>0} ;
    2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle u=x-y+2z} , ;
    3. , , ;

Typical questions for seminar classes (labs) within this section

  1. Let us consider 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 .
  2. Find the largest possible value of directional derivative at point of function .
  3. Find maxima and minima of functions given implicitly by the equations:
    1. , ;
    2. .
  4. Find maxima and minima of functions subject to constraints:
    1. , ;
    2. , , , .

Test questions for final assessment in this section

  1. Find all points where the differential of a function is equal to zero.
  2. Show that function satisfies the equation .
  3. Find maxima and minima of function under condition that . Find the maximum and minimum value of a function
  4. on a domain given by inequality ;

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: where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle D=\left\{(x;y)\left|x^2+y^2\leq9,\,x^2+(y+4)^2\geq25\right.\right\}} .
  2. Represent integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle I=\displaystyle\iiint\limits_Df(x;y;z)\,dx\,dy\,dz} as iterated integrals with all possible (i.e. 6) orders of integration; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle D} is bounded by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x=a} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y=\sqrt{ax}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z=0} , .
  3. Find line integrals of a scalar fields Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\int\limits_{\Gamma}(x+y)\,ds} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Gamma} is boundary of a triangle with vertices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle (0;0)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle (1;0)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle (0;1)} .

Typical questions for seminar classes (labs) within this section

  1. Change order of integration in the iterated integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \int\limits_0^{\sqrt2}dy\int\limits_y^{\sqrt{4-y^2}}f(x;y)\,dx} .
  2. Find the volume of a solid given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle 0\leq z\leq x^2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x+y\leq 5} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x-2y\geq2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y\geq0} .
  3. Change into polar coordinates and rewrite the integral as a single integral: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\iint\limits_Gf\left(\sqrt{x^2+y^2}\right)\,dx\,dy} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle A} and finishes at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle B} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\int\limits_{\Gamma}\left(x^4+4xy^3\right)\,dx +\left(6x^2y^2-5y^4\right)\,dy} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle A(-2;-1)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle B(0;3)} ;

Test questions for final assessment in this section

  1. Domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle G} is bounded by lines Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y=2x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y=x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y=2} . Rewrite integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \iint\limits_Gf(x)\,dx\,dy} as a single integral.
  2. Represent the integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\iint\limits_Gf(x;y)\,dx\,dy} as iterated integrals with different order of integration in polar coordinates if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle G=\left\{(x;y)\left|a^2\leq x^2+y^2\leq 4a^2;\,|x|-y\geq0\right.\right\}} .
  3. Find the integral making an appropriate substitution: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\iiint\limits_G\left(x^2-y^2\right)\left(z+x^2-y^2\right)\,dx\,dy\,dz} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\iint\limits_S(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle S} is the outer surface of a tetrahedron Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac xa+\frac yb+\frac zc\leq1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle x\geq0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle y\geq0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z\geq0} ;

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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sum\limits_{n=1}^{\infty}e^{-n\left(x^2+2\sin x\right)}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta_1=(0;1]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta_2=[1;+\infty)} ;
  2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sum\limits_{n=1}^{\infty}\frac{\sqrt{nx^3}}{x^2+n^2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta_1=(0;1)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta_2=(1;+\infty)}
  3. Show that sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f_n(x)=nx\left(1-x\right)^n} converges non-uniformly on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle [0;1]} to a continuous function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x)} , but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim\limits_{n\rightarrow+\infty}\int\limits_0^1f_n(x)\,dx=\lim\limits_{n\rightarrow+\infty}\int\limits_0^1f(x)\,dx} .
  4. Decompose the following function determined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle [-\pi;\pi]} into Fourier series using the standard trigonometric system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \left.\left\{1;\cos kx;\sin kx\right\}\right|_{k=1}^{\infty}} . Draw the graph of the sum of Fourier series obtained. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x)} on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle [-\pi;\pi]}
    1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x+\pi)=f(x)} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a_{2k-1}=b_{2k-1}=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle k\in\mathbb{N}} ;
    2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x+\pi)=-f(x)} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a_0=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a_{2k}=b_{2k}=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle k\in\mathbb{N}} .

Typical questions for seminar classes (labs) within this section

  1. Show that sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f_n(x)=nx\left(1-x^2\right)^n} converges on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle [0;1]} to a continuous function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x)} , and at that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim\limits_{n\rightarrow+\infty}\int\limits_0^1f_n(x)\,dx\neq\lim\limits_{n\rightarrow+\infty}\int\limits_0^1f(x)\,dx} .
  2. Show that sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f_n(x)=x^3+\frac1n\sin\left(nx+\frac{n\pi}2\right)} converges uniformly on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathbb{R}} , but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \left(\lim\limits_{n\rightarrow+\infty}f_n(x)\right)'\neq\lim\limits_{n\rightarrow+\infty}f'_n(x)} .
  3. Decompose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \cos\alpha x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \alpha\notin\mathbb{Z}} into Fourier series on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle [-\pi;\pi]} . Using this decomposition prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \cot y=\frac1y+\sum\limits_{k=1}^{\infty}\frac{2y}{y^2-\pi^2k^2}} .
  4. Function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x)} is absolutely integrable on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle [0;\pi]} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(\pi-x)=f(x)} . Prove that
    1. if it is decomposed into Fourier series of sines then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle b_{2k}=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle k\in\mathbb{N}} ;
    2. if it is decomposed into Fourier series of cosines then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a_{2k-1}=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle k\in\mathbb{N}} .
  5. ## Decompose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sigma_1=\sum\limits_{k=1}^{\infty}\frac{\sin^2k\alpha}{k^2}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sigma_2=\sum\limits_{k=1}^{\infty}\frac{\cos^2k\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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sum\limits_{n=1}^{\infty}\frac{xn+\sqrt n}{n+x}\ln\left(1+\frac x{n\sqrt n}\right)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta_1=(0;1)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta_2=(1;+\infty)} ;
  2. Show that sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f_n(x)=\frac{\sin nx}{\sqrt n}} converges uniformly on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathbb{R}} to a differentiable function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x)} , and at that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\int\limits_0^1\left(\lim\limits_{\alpha\to0}\frac{2x\alpha^2}{\left(\alpha^2+x^2\right)^2}\right)\,dx= \lim\limits_{\alpha\to0}\int\limits_0^1\frac{2x\alpha^2}{\left(\alpha^2+x^2\right)^2}\,dx} .
  2. Differentiating the integrals with respect to parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \varphi} , find it: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle I(\alpha)=\int\limits_0^{\pi/2}\ln\left(\alpha^2-\sin^2\varphi\right)\,d\varphi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \alpha>1} .
  3. Prove that the following integral converges uniformly on the indicated set. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\int\limits_0^{+\infty}e^{-\alpha x}\cos2x\,dx} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta=[1;+\infty)} ;
  4. It is known that Dirichlet’s integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \int\limits_0^{+\infty}\frac{\sin x}x\,dx} is equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac\pi2} . Find the values of the following integrals using Dirichlet’s integral
    1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \int\limits_0^{+\infty}\frac\sin{\alpha x}x\,dx} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \alpha\neq0} ;
    2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\int\limits_0^1\left(\int\limits_0^1f(x,\alpha)\,d\alpha\right)\,dx= \int\limits_0^1\left(\int\limits_0^1f(x,\alpha)\,dx\right)\,d\alpha} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x;\alpha)=\frac{\alpha-x}{(\alpha+x)^3}} .
  2. Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Phi'(\alpha)} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Phi(\alpha)=\int\limits_1^2\frac{e^{\alpha x^2}}x\,dx} .
  3. Differentiating the integral with respect to parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \alpha} , find it: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle I(\alpha)=\int\limits_0^\pi\frac1{\cos x} \ln\frac{1+\alpha\cos x}{1-\alpha\cos x}\,dx} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle |\alpha|<1} .
  4. Find Fourier transform of the following functions:
    1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x)=\begin{cases}1,&|x|\leq1,\\0,&|x|>1;\end{cases}}
  5. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \widehat{f}(y)} be Fourier transform of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x)} . Prove that Fourier transform of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle e^{i\alpha x}f(x)} is equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \widehat{f}(y-\alpha)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \alpha\in\mathbb{R}} .

Test questions for final assessment in this section

  1. Find out if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\int\limits_0^1\left(\int\limits_0^1f(x,\alpha)\,d\alpha\right)\,dx= \int\limits_0^1\left(\int\limits_0^1f(x,\alpha)\,dx\right)\,d\alpha} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x;\alpha)=\frac{\alpha^2-x^2}{\left(\alpha^2+x^2\right)^2}} .
  2. Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Phi'(\alpha)} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \displaystyle\int\limits_{-\infty}^{+\infty}\frac{\cos\alpha x}{4+x^2}\,dx} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \Delta=\mathbb{R}} ;
  4. Find Fourier integral for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x)=\begin{cases}1,&|x|\leq\tau,\\0,&|x|>\tau;\end{cases}}
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