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# Find maxima and minima of function $u=2x^2+12xy+y^2$ under condition that $x^2+4y^2=25$. |
# Find maxima and minima of function $u=2x^2+12xy+y^2$ under condition that $x^2+4y^2=25$. |
||
# $u=\left(y^2-x^2\right)e^{1-x^2+y^2}$ on a domain given by inequality \quad $x^2+y^2\leq4$; |
# $u=\left(y^2-x^2\right)e^{1-x^2+y^2}$ on a domain given by inequality \quad $x^2+y^2\leq4$; |
||
| + | === 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? ==== |
||
| + | {| class="wikitable" |
||
| + | |+ |
||
| + | |- |
||
| + | ! 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 ==== |
||
| + | # Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: $\iint\limits_Df(x;y)\,dx\,dy$ where $D=\left\{(x;y)\left|x^2+y^2\leq9,\,x^2+(y+4)^2\geq25\right.\right\}$. |
||
| + | # Represent integral $I=\displaystyle\iiint\limits_Df(x;y;z)\,dx\,dy\,dz$ as iterated integrals with all possible (i.e. 6) orders of integration; $D$ is bounded by $x=0$, $x=a$, $y=0$, $y=\sqrt{ax}$, $z=0$, $z=x+y$. |
||
| + | # 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 ==== |
||
| + | # Change order of integration in the iterated integral |
||
| + | # Find the volume of a solid given by $0\leq z\leq x^2$, $x+y\leq 5$, $x-2y\geq2$, $y\geq0$. |
||
| + | # Change into polar coordinates and rewrite the integral as a single integral: |
||
| + | # Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at $A$ and finishes at $B$: $\displaystyle\int\limits_{\Gamma}\left(x^4+4xy^3\right)\,dx +\left(6x^2y^2-5y^4\right)\,dy$, $A(-2;-1)$, $B(0;3)$; |
||
| + | |||
| + | ==== Tasks for midterm assessment within this section ==== |
||
| + | |||
| + | |||
| + | ==== Test questions for final assessment in this section ==== |
||
| + | # Domain $G$ is bounded by lines $y=2x$, $y=x$ and $y=2$. Rewrite integral $\iint\limits_Gf(x)\,dx\,dy$ as a single integral. |
||
| + | # Represent the integral $\displaystyle\iint\limits_Gf(x;y)\,dx\,dy$ as iterated integrals with different order of integration in polar coordinates if $G=\left\{(x;y)\left|a^2\leq x^2+y^2\leq 4a^2;\,|x|-y\geq0\right.\right\}$. |
||
| + | # Find the integral making an appropriate substitution: |
||
| + | # 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 $\frac xa+\frac yb+\frac zc\leq1$, $x\geq0$, $y\geq0$, $z\geq0$; |
||
Revision as of 15:51, 2 December 2021
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
| Type | Points |
|---|---|
| Test 1 | 10 |
| Midterm | 25 |
| Test 2 | 10 |
| Participation | 5 |
Grades 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
- Find $\lim\limits_{x\to0}\lim\limits_{y\to0}u(x;y)$, $\lim\limits_{y\to0}\lim\limits_{x\to0}u(x;y)$ and $\lim\limits_{(x;y)\to(0;0)}u(x;y)$ if $u(x;y)=\frac{x^2y+xy^2}{x^2-xy+y^2}$.
- Find the differential of a function: (a)~$u(x;y)=\ln\left(x+\sqrt{x^2+y^2}\right)$; \; (b)~$u(x;y)=\ln\sin\frac{x+1}{\sqrt y}$.
- 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 $N(1;1;-2)$.
- Find maxima and minima of a function subject to a constraint (or several constraints): \begin{enumerate}
- $u=x^2y^3z^4$, \quad $2x+3y+4z=18$, $x>0$, $y>0$, $z>0$;
- $u=x-y+2z$, \quad $x^2+y^2+2z^2=16$;
- $u=\sum\limits_{i=1}^ka_ix_i^2$, \quad $\sum\limits_{i=1}^kx_i=1$, $a_i>0$;
Typical questions for seminar classes (labs) within this section
- Let us consider $u(x;y)=\begin{cases}1,&x=y^2,\\0,&x\neq y^2.\end{
Tasks for midterm assessment within this section
Test questions for final assessment in this section
- 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.
- Show that function $\varphi=f\left(\frac xy;x^2+y-z^2\right)$ satisfies the equation $2xz\varphi_x+2yz\varphi_y+\left(2x^2+y\right)\varphi_z=0$.
- Find maxima and minima of function $u=2x^2+12xy+y^2$ under condition that $x^2+4y^2=25$.
- $u=\left(y^2-x^2\right)e^{1-x^2+y^2}$ on a domain given by inequality \quad $x^2+y^2\leq4$;
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
- Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: $\iint\limits_Df(x;y)\,dx\,dy$ where $D=\left\{(x;y)\left|x^2+y^2\leq9,\,x^2+(y+4)^2\geq25\right.\right\}$.
- Represent integral $I=\displaystyle\iiint\limits_Df(x;y;z)\,dx\,dy\,dz$ as iterated integrals with all possible (i.e. 6) orders of integration; $D$ is bounded by $x=0$, $x=a$, $y=0$, $y=\sqrt{ax}$, $z=0$, $z=x+y$.
- 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
- Change order of integration in the iterated integral
- Find the volume of a solid given by $0\leq z\leq x^2$, $x+y\leq 5$, $x-2y\geq2$, $y\geq0$.
- Change into polar coordinates and rewrite the integral as a single integral:
- Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at $A$ and finishes at $B$: $\displaystyle\int\limits_{\Gamma}\left(x^4+4xy^3\right)\,dx +\left(6x^2y^2-5y^4\right)\,dy$, $A(-2;-1)$, $B(0;3)$;
Tasks for midterm assessment within this section
Test questions for final assessment in this section
- Domain $G$ is bounded by lines $y=2x$, $y=x$ and $y=2$. Rewrite integral $\iint\limits_Gf(x)\,dx\,dy$ as a single integral.
- Represent the integral $\displaystyle\iint\limits_Gf(x;y)\,dx\,dy$ as iterated integrals with different order of integration in polar coordinates if $G=\left\{(x;y)\left|a^2\leq x^2+y^2\leq 4a^2;\,|x|-y\geq0\right.\right\}$.
- Find the integral making an appropriate substitution:
- 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 $\frac xa+\frac yb+\frac zc\leq1$, $x\geq0$, $y\geq0$, $z\geq0$;