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MathematicalAnalysis II

• Course name: MathematicalAnalysis II
• Code discipline:
• Subject area: Multivariate calculus: derivatives, differentials, maxima and minima, Multivariate integration, Functional series. Fourier series, Integrals with parameters

Course Topics

Course Sections and Topics

Section	Topics within the section
Differential Analysis of Functions of Several Variables	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
Integration of Functions of Several Variables	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
Uniform Convergence of Functional Series. Fourier Series	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.
Integrals with Parameter(s)	Definite integrals with parameters Improper integrals with parameters. Uniform convergence Properties of uniformly convergent integrals Beta-function and gamma-function Fourier transform

Intended Learning Outcomes (ILOs)

What is the main 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.

ILOs defined at three levels

Level 1: What concepts should a student know/remember/explain?

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

Level 2: What basic practical skills should a student be able to perform?

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

• 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

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

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

Grading

Course grading range

Grade	Range	Description of performance
A. Excellent	85-100	-
B. Good	65-84	-
C. Satisfactory	45-64	-
D. Poor	0-44	-

Course activities and grading breakdown

Activity Type	Percentage of the overall course grade
Test 1	10
Midterm	25
Test 2	10
Participation	5
Final exam	50

Resources, literature and reference materials

Open access resources

• 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

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section

Learning Activities	Section 1	Section 2	Section 3	Section 4
Homework and group projects	1	1	1	1
Midterm evaluation	1	1	1	1
Testing (written or computer based)	1	1	1	1
Discussions	1	1	1	1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type	Content	Is Graded?
Question	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 ${\textstyle u(x;y)={\frac {x^{2}y+xy^{2}}{x^{2}-xy+y^{2}}}}$ .	1
Question	Find the differential of a function: (a) ${\textstyle u(x;y)=\ln \left(x+{\sqrt {x^{2}+y^{2}}}\right)}$ ; (b) ${\textstyle u(x;y)=\ln \sin {\frac {x+1}{\sqrt {y}}}}$ .	1
Question	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 ${\textstyle N(1;1;-2)}$ .	1
Question	Find maxima and minima of a function subject to a constraint (or several constraints): ${\textstyle u=x^{2}y^{3}z^{4}}$ , ${\textstyle 2x+3y+4z=18}$ , ${\textstyle x>0}$ , ${\textstyle y>0}$ , ${\textstyle z>0}$ ; ${\textstyle u=x-y+2z}$ , ${\textstyle x^{2}+y^{2}+2z^{2}=16}$ ; ${\textstyle u=\sum \limits _{i=1}^{k}a_{i}x_{i}^{2}}$ , ${\textstyle \sum \limits _{i=1}^{k}x_{i}=1}$ , ${\textstyle a_{i}>0}$ ;	1
Question	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 ${\textstyle (x;y)\to (0;0)}$ .	0
Question	Find the largest possible value of directional derivative at point ${\textstyle M(1;-2;-3)}$ of function ${\textstyle f=\ln xyz}$ .	0
Question	Find maxima and minima of functions ${\textstyle u(x,y)}$ given implicitly by the equations: ${\textstyle x^{2}+y^{2}+u^{2}-4x-6y-4u+8=0}$ , ${\textstyle u>2}$ ; ${\textstyle x^{3}-y^{2}+u^{2}-3x+4y+u-8=0}$ .	0
Question	Find maxima and minima of functions subject to constraints: ${\textstyle u=xy^{2}}$ , ${\textstyle x+2y-1=0}$ ; ${\textstyle u=xy+yz}$ , ${\textstyle x^{2}+y^{2}=2}$ , ${\textstyle y+z=2}$ , ${\textstyle y>0}$ .	0

Section 2

Activity Type	Content	Is Graded?
Question	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 ${\textstyle D=\left\{(x;y)\left|x^{2}+y^{2}\leq 9,\,x^{2}+(y+4)^{2}\geq 25\right.\right\}}$ .	1
Question	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 ${\textstyle x=0}$ , ${\textstyle x=a}$ , ${\textstyle y=0}$ , ${\textstyle y={\sqrt {ax}}}$ , ${\textstyle z=0}$ , ${\textstyle z=x+y}$ .	1
Question	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 ${\textstyle (0;1)}$ .	1
Question	Change order of integration in the iterated integral ${\textstyle \int \limits _{0}^{\sqrt {2}}dy\int \limits _{y}^{\sqrt {4-y^{2}}}f(x;y)\,dx}$ .	0
Question	Find the volume of a solid given by ${\textstyle 0\leq z\leq x^{2}}$ , ${\textstyle x+y\leq 5}$ , ${\textstyle x-2y\geq 2}$ , ${\textstyle y\geq 0}$ .	0
Question	Change into polar coordinates and rewrite the integral as a single integral: ${\textstyle \displaystyle \iint \limits _{G}f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx\,dy}$ , ${\textstyle G=\left\{(x;y)\left|x^{2}+y^{2}\leq x;\,x^{2}+y^{2}\leq y\right.\right\}}$ .	0
Question	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 ${\textstyle B}$ : ${\textstyle \displaystyle \int \limits _{\Gamma }\left(x^{4}+4xy^{3}\right)\,dx+\left(6x^{2}y^{2}-5y^{4}\right)\,dy}$ , ${\textstyle A(-2;-1)}$ , ${\textstyle B(0;3)}$ ;	0

Section 3

Activity Type	Content	Is Graded?
Question	Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. ${\textstyle \sum \limits _{n=1}^{\infty }e^{-n\left(x^{2}+2\sin x\right)}}$ , ${\textstyle \Delta _{1}=(0;1]}$ , ${\textstyle \Delta _{2}=[1;+\infty )}$ ;	1
Question	${\textstyle \sum \limits _{n=1}^{\infty }{\frac {\sqrt {nx^{3}}}{x^{2}+n^{2}}}}$ , ${\textstyle \Delta _{1}=(0;1)}$ , ${\textstyle \Delta _{2}=(1;+\infty )}$	1
Question	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 ${\textstyle \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x)\,dx=\lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x)\,dx}$ .	1
Question	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. ${\textstyle f(x)={\begin{cases}1,\;0\leq x\leq \pi ,\\0,\;-\pi \leq x<0.\end{cases}}}$	1
Question	Prove that if for an absolutely integrable function ${\textstyle f(x)}$ on ${\textstyle [-\pi ;\pi ]}$ ${\textstyle f(x+\pi )=f(x)}$ then ${\textstyle a_{2k-1}=b_{2k-1}=0}$ , ${\textstyle k\in \mathbb {N} }$ ; ${\textstyle f(x+\pi )=-f(x)}$ then ${\textstyle a_{0}=0}$ , ${\textstyle a_{2k}=b_{2k}=0}$ , ${\textstyle k\in \mathbb {N} }$ .	1
Question	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 ${\textstyle \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x)\,dx\neq \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x)\,dx}$ .	0
Question	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 ${\textstyle \left(\lim \limits _{n\rightarrow +\infty }f_{n}(x)\right)'\neq \lim \limits _{n\rightarrow +\infty }f'_{n}(x)}$ .	0
Question	Decompose ${\textstyle \cos \alpha x}$ , ${\textstyle \alpha \notin \mathbb {Z} }$ into Fourier series on ${\textstyle [-\pi ;\pi ]}$ . Using this decomposition prove that ${\textstyle \cot y={\frac {1}{y}}+\sum \limits _{k=1}^{\infty }{\frac {2y}{y^{2}-\pi ^{2}k^{2}}}}$ .	0
Question	Function ${\textstyle f(x)}$ is absolutely integrable on ${\textstyle [0;\pi ]}$ , and ${\textstyle f(\pi -x)=f(x)}$ . Prove that if it is decomposed into Fourier series of sines then ${\textstyle b_{2k}=0}$ , ${\textstyle k\in \mathbb {N} }$ ; if it is decomposed into Fourier series of cosines then ${\textstyle a_{2k-1}=0}$ , ${\textstyle k\in \mathbb {N} }$ .	0
Question	## Decompose ${\textstyle f(x)={\begin{cases}1,\;|x|<\alpha ,\\0,\;\alpha \leqslant |x|<\pi \end{cases}}}$ into Fourier series using the standard trigonometric system. Using Parseval’s identity find ${\textstyle \sigma _{1}=\sum \limits _{k=1}^{\infty }{\frac {\sin ^{2}k\alpha }{k^{2}}}}$ and ${\textstyle \sigma _{2}=\sum \limits _{k=1}^{\infty }{\frac {\cos ^{2}k\alpha }{k^{2}}}}$ .	0

Section 4

Activity Type	Content	Is Graded?
Question	Find out if ${\textstyle \displaystyle \int \limits _{0}^{1}\left(\lim \limits _{\alpha \to 0}{\frac {2x\alpha ^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}\right)\,dx=\lim \limits _{\alpha \to 0}\int \limits _{0}^{1}{\frac {2x\alpha ^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}\,dx}$ .	1
Question	Differentiating the integrals with respect to parameter ${\textstyle \varphi }$ , find it: ${\textstyle I(\alpha )=\int \limits _{0}^{\pi /2}\ln \left(\alpha ^{2}-\sin ^{2}\varphi \right)\,d\varphi }$ , ${\textstyle \alpha >1}$ .	1
Question	Prove that the following integral converges uniformly on the indicated set. ${\textstyle \displaystyle \int \limits _{0}^{+\infty }e^{-\alpha x}\cos 2x\,dx}$ , ${\textstyle \Delta =[1;+\infty )}$ ;	1
Question	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 ${\textstyle \int \limits _{0}^{+\infty }{\frac {\sin }{\alpha x}}x\,dx}$ , ${\textstyle \alpha \neq 0}$ ; ${\textstyle \int \limits _{0}^{+\infty }{\frac {\sin x-x\cos x}{x^{3}}}\,dx}$ .	1
Question	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 ${\textstyle f(x;\alpha )={\frac {\alpha -x}{(\alpha +x)^{3}}}}$ .	0
Question	Find ${\textstyle \Phi '(\alpha )}$ if ${\textstyle \Phi (\alpha )=\int \limits _{1}^{2}{\frac {e^{\alpha x^{2}}}{x}}\,dx}$ .	0
Question	Differentiating the integral with respect to parameter ${\textstyle \alpha }$ , find it: ${\textstyle I(\alpha )=\int \limits _{0}^{\pi }{\frac {1}{\cos x}}\ln {\frac {1+\alpha \cos x}{1-\alpha \cos x}}\,dx}$ , ${\textstyle |\alpha |<1}$ .	0
Question	Find Fourier transform of the following functions: ${\textstyle f(x)={\begin{cases}1,&|x|\leq 1,\\0,&|x|>1;\end{cases}}}$	0
Question	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 ${\textstyle {\widehat {f}}(y-\alpha )}$ , ${\textstyle \alpha \in \mathbb {R} }$ .	0

Final assessment

Section 1

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

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

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 4

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 ${\textstyle f(x)={\begin{cases}1,&|x|\leq \tau ,\\0,&|x|>\tau ;\end{cases}}}$

The retake exam

Section 1

Section 2

Section 3

Section 4