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Calculus II

• Course name: Calculus II
• Course number: XYZ

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

• 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

Type	Points
Test 1	10
Midterm	25
Test 2	10
Participation	5

Grades range

Course grading range

Grade	Points
A	[85, 100]
B	[65, 84]
C	[45, 64]
D	[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:

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?

Form	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	0

Typical questions for ongoing performance evaluation within this section

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

Typical questions for seminar classes (labs) within this section

1. Let us consider $u(x;y)={\begin{cases}1,&x=y^{2},\\0,&x\neq y^{2}.\end{cases}}$

Test questions for final assessment in this section

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

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?

Form	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	0

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

Typical questions for seminar classes (labs) within this section

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

Test questions for final assessment in this section

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

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?

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

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.
2. $\sum \limits _{n=1}^{\infty }{\frac {\sqrt {nx^{3}}}{x^{2}+n^{2}}}$, $\Delta _{1}=(0;1)$, $\Delta _{2}=(1;+\infty )$
3. Show that sequence $f_{n}(x)=nx\left(1-x\right)^{n}$ converges non-uniformly on $[0;1]$ to a continuous function $f(x)$, but $\lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x)\,dx=\lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x)\,dx$.
4. Decompose the following function determined on $[-\pi ;\pi ]$ into Fourier series using the standard trigonometric system $\left.\left\{1;\cos kx;\sin kx\right\}\right|_{k=1}^{\infty }$. Draw the graph of the sum of Fourier series obtained.

Typical questions for seminar classes (labs) within this section

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

Test questions for final assessment in this section

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