raw/raw_bsc__mathematical_analysis_ii.s23.md

BSc: Mathematical Analysis II.s23
=================================






Contents
--------


* [1 Mathematical Analysis II](#Mathematical_Analysis_II)
	+ [1.1 Short Description](#Short_Description)
	+ [1.2 Course Topics](#Course_Topics)
	+ [1.3 Intended Learning Outcomes (ILOs)](#Intended_Learning_Outcomes_.28ILOs.29)
		- [1.3.1 What is the main purpose of this course?](#What_is_the_main_purpose_of_this_course.3F)
		- [1.3.2 ILOs defined at three levels](#ILOs_defined_at_three_levels)
			* [1.3.2.1 Level 1: What concepts should a student know/remember/explain?](#Level_1:_What_concepts_should_a_student_know.2Fremember.2Fexplain.3F)
			* [1.3.2.2 Level 2: What basic practical skills should a student be able to perform?](#Level_2:_What_basic_practical_skills_should_a_student_be_able_to_perform.3F)
			* [1.3.2.3 Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?](#Level_3:_What_complex_comprehensive_skills_should_a_student_be_able_to_apply_in_real-life_scenarios.3F)
	+ [1.4 Grading](#Grading)
		- [1.4.1 Course grading range](#Course_grading_range)
		- [1.4.2 Course activities and grading breakdown](#Course_activities_and_grading_breakdown)
		- [1.4.3 Recommendations for students on how to succeed in the course](#Recommendations_for_students_on_how_to_succeed_in_the_course)
	+ [1.5 Resources, literature and reference materials](#Resources.2C_literature_and_reference_materials)
		- [1.5.1 Open access resources](#Open_access_resources)
	+ [1.6 Activities and Teaching Methods](#Activities_and_Teaching_Methods)
	+ [1.7 Formative Assessment and Course Activities](#Formative_Assessment_and_Course_Activities)
		- [1.7.1 Ongoing performance assessment](#Ongoing_performance_assessment)
			* [1.7.1.1 Section 1](#Section_1)
			* [1.7.1.2 Section 2](#Section_2)
			* [1.7.1.3 Section 3](#Section_3)
			* [1.7.1.4 Section 4](#Section_4)
		- [1.7.2 Final assessment](#Final_assessment)
			* [1.7.2.1 Section 1](#Section_1_2)
			* [1.7.2.2 Section 2](#Section_2_2)
			* [1.7.2.3 Section 3](#Section_3_2)
			* [1.7.2.4 Section 4](#Section_4_2)
		- [1.7.3 The retake exam](#The_retake_exam)



Mathematical Analysis II
========================


* **Course name**: Mathematical Analysis II
* **Code discipline**: CSE203
* **Subject area**: Math


Short Description
-----------------


* Series: convergence, approximation
* Multivariate calculus: derivatives, differentials, maxima and minima
* Multivariate integration
* Basics of vector analysis


Course Topics
-------------




Course Sections and Topics
| Section | Topics within the section
 |
| --- | --- |
| Infinite Series | 1. The Sum of an Infinite Series
2. The Comparison Test
3. The Integral and Ratio Tests
4. Alternating Series
5. Power Series
6. Taylor's Formula
 |
| Partial Differentiation | 1. Limits of functions of several variables
2. Introduction to Partial Derivatives
3. The Chain Rule
4. Gradients
5. Level Surfaces and Implicit Differentiation
6. Maximas and Minimas
7. Constrained Extrema and Lagrange Multipliers
 |
| Multiple Integration | 1. The Double Integral and Iterated Integral
2. The Double Integral over General Region
3. Integrals in Polar coordinates, Substitutions in the double integrals
4. Integrals in Cylindrical and Spherical Coordinates
5. Applications of the Double and Triple Integrals
 |
| Vector Analysis | 1. Line Integrals, Path Independence
2. Exact Differentials
3. Green’s Theorem
4. Circulation and Stoke’s Theorem
5. Flux and Divergence Theorem
 |


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


We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.



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


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



* know how to find minima and maxima of a function subject to a constraint
* know how to represent double integrals as iterated integrals and vice versa
* know what the length of a curve and the area of a surface is


#### 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 ...



* find partial and directional derivatives of functions of several variables;
* find maxima and minima for a function of several variables
* use Fubini theorem for calculating multiple integrals
* calculate line and path integrals


#### 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 infinite series


Grading
-------


### Course grading range





| Grade | Range | Description of performance
 |
| --- | --- | --- |
| A. Excellent | 90-100 | -
 |
| B. Good | 75-89 | -
 |
| C. Satisfactory | 60-74 | -
 |
| D. Fail | 0-59 | -
 |


### Course activities and grading breakdown





| Activity Type | Percentage of the overall course grade
 |
| --- | --- |
| Midterm | 20
 |
| Quizzes | 28 (2 for each)
 |
| Final exam | 50
 |
| In-class participation | 7 (including 5 extras)
 |


### Recommendations for students on how to succeed in the course


* Participation is important. Attending lectures is the key to success in this course.
* Review lecture materials before classes to do well.
* Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.


Resources, literature and reference materials
---------------------------------------------


### Open access resources


* Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985 [link](https://www.cds.caltech.edu/~marsden/volume/Calculus/)
* Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson


Activities and Teaching Methods
-------------------------------




Teaching and Learning Methods within each section
| Teaching Techniques | Section 1 | Section 2 | Section 3 | Section 4
 |
| --- | --- | --- | --- | --- |
| Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) | 0 | 0 | 0 | 0
 |
| Project-based learning (students work on a project) | 0 | 0 | 0 | 0
 |
| Modular learning (facilitated self-study) | 0 | 0 | 0 | 0
 |
| Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level) | 1 | 1 | 1 | 1
 |
| Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them) | 0 | 0 | 0 | 0
 |
| Business game (learn by playing a game that incorporates the principles of the material covered within the course) | 0 | 0 | 0 | 0
 |
| Inquiry-based learning | 0 | 0 | 0 | 0
 |
| Just-in-time teaching | 0 | 0 | 0 | 0
 |
| Process oriented guided inquiry learning (POGIL) | 0 | 0 | 0 | 0
 |
| Studio-based learning | 0 | 0 | 0 | 0
 |
| Universal design for learning | 0 | 0 | 0 | 0
 |
| Task-based learning | 0 | 0 | 0 | 0
 |




Activities within each section
| Learning Activities | Section 1 | Section 2 | Section 3 | Section 4
 |
| --- | --- | --- | --- | --- |
| Lectures | 1 | 1 | 1 | 1
 |
| Interactive Lectures | 1 | 1 | 1 | 1
 |
| Lab exercises | 1 | 1 | 1 | 1
 |
| Experiments | 0 | 0 | 0 | 0
 |
| Modeling | 0 | 0 | 0 | 0
 |
| Cases studies | 0 | 0 | 0 | 0
 |
| Development of individual parts of software product code | 0 | 0 | 0 | 0
 |
| Individual Projects | 0 | 0 | 0 | 0
 |
| Group projects | 0 | 0 | 0 | 0
 |
| Flipped classroom | 0 | 0 | 0 | 0
 |
| Quizzes (written or computer based) | 1 | 1 | 1 | 1
 |
| Peer Review | 0 | 0 | 0 | 0
 |
| Discussions | 1 | 1 | 1 | 1
 |
| Presentations by students | 0 | 0 | 0 | 0
 |
| Written reports | 0 | 0 | 0 | 0
 |
| Simulations and role-plays | 0 | 0 | 0 | 0
 |
| Essays | 0 | 0 | 0 | 0
 |
| Oral Reports | 0 | 0 | 0 | 0
 |


Formative Assessment and Course Activities
------------------------------------------


### Ongoing performance assessment


#### Section 1


1. Derive the Maclaurin expansion for 



f
(
x
)
=



1
+

e

−
2
x




3





{\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}

![{\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14bfa69ed7bcfd7de396a3360c031c64292828b3) up to 



o

(

x

3


)



{\textstyle o\left(x^{3}\right)}

![{\textstyle o\left(x^{3}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e23f841ca36ac3c4d7ccc3920dbe12d69f5b304).
2. Find 




lim

x
→
0



lim

y
→
0


u
(
x
;
y
)


{\textstyle \lim \limits \_{x\to 0}\lim \limits \_{y\to 0}u(x;y)}

![{\textstyle \lim \limits _{x\to 0}\lim \limits _{y\to 0}u(x;y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41fcf6ef0f4de155d38dc7b27a24cb3be9c95abc), 




lim

y
→
0



lim

x
→
0


u
(
x
;
y
)


{\textstyle \lim \limits \_{y\to 0}\lim \limits \_{x\to 0}u(x;y)}

![{\textstyle \lim \limits _{y\to 0}\lim \limits _{x\to 0}u(x;y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30b9ab013043fea23bc9bc3584592aa95379e778) and 




lim

(
x
;
y
)
→
(
0
;
0
)


u
(
x
;
y
)


{\textstyle \lim \limits \_{(x;y)\to (0;0)}u(x;y)}

![{\textstyle \lim \limits _{(x;y)\to (0;0)}u(x;y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f181440f0fe3a34a0575a79be4be5e8e6aa37957) if 



u
(
x
;
y
)
=




x

2


y
+
x

y

2





x

2


−
x
y
+

y

2







{\textstyle u(x;y)={\frac {x^{2}y+xy^{2}}{x^{2}-xy+y^{2}}}}

![{\textstyle u(x;y)={\frac {x^{2}y+xy^{2}}{x^{2}-xy+y^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7a5e54b99cd1e3fd00b1f8ae13ab79a21e5f56).


#### Section 2


1. Find the differential of a function: (a) 



u
(
x
;
y
)
=
ln
⁡

(

x
+



x

2


+

y

2





)



{\textstyle u(x;y)=\ln \left(x+{\sqrt {x^{2}+y^{2}}}\right)}

![{\textstyle u(x;y)=\ln \left(x+{\sqrt {x^{2}+y^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cc7b02dad3ea12644b4ecadf08fcaa6405d4529); (b) 



u
(
x
;
y
)
=
ln
⁡
sin
⁡



x
+
1


y





{\textstyle u(x;y)=\ln \sin {\frac {x+1}{\sqrt {y}}}}

![{\textstyle u(x;y)=\ln \sin {\frac {x+1}{\sqrt {y}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c25c88f1c0c5fda11150ba0fe2212b28a40ecc7).
2. Find the differential of 



u
(
x
;
y
)


{\textstyle u(x;y)}

![{\textstyle u(x;y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feeabc2f3e55a2a2f72b6ec04a08c1315c08699e) given implicitly by an equation 




x

3


+
2

y

3


+

u

3


−
3
x
y
u
+
2
y
−
3
=
0


{\textstyle x^{3}+2y^{3}+u^{3}-3xyu+2y-3=0}

![{\textstyle x^{3}+2y^{3}+u^{3}-3xyu+2y-3=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/911701538f15cf656e0dd261ad813f9c8a2c93f9) at points 



M
(
1
;
1
;
1
)


{\textstyle M(1;1;1)}

![{\textstyle M(1;1;1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97b7d6ccd5c6763a4d4ea7e712c393483057239d) and 



N
(
1
;
1
;
−
2
)


{\textstyle N(1;1;-2)}

![{\textstyle N(1;1;-2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c66cfa579443b4e19dcd7f9348d1ae9dfce8d4).
3. Find maxima and minima of a function subject to a constraint (or several constraints):
	1. u
	=
	
	x
	
	2
	
	
	
	y
	
	3
	
	
	
	z
	
	4
	
	
	
	
	{\textstyle u=x^{2}y^{3}z^{4}}
	
	![{\textstyle u=x^{2}y^{3}z^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9178002e5b4a33a04c2d25eb41d3311427efe6d), 
	
	
	
	2
	x
	+
	3
	y
	+
	4
	z
	=
	18
	
	
	{\textstyle 2x+3y+4z=18}
	
	![{\textstyle 2x+3y+4z=18}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a5abaa6c5cd03742a1c4a119e9c686645f2018), 
	
	
	
	x
	>
	0
	
	
	{\textstyle x>0}
	
	![{\textstyle x>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd6212b5a778bded18868fc5cd19eb3b15844a0), 
	
	
	
	y
	>
	0
	
	
	{\textstyle y>0}
	
	![{\textstyle y>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bad11e7747271c5d6ebd668b512129802597fc5c), 
	
	
	
	z
	>
	0
	
	
	{\textstyle z>0}
	
	![{\textstyle z>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3045a2beda2df2d38cfc9478a0eea2ddb453540);
	2. u
	=
	x
	−
	y
	+
	2
	z
	
	
	{\textstyle u=x-y+2z}
	
	![{\textstyle u=x-y+2z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/193769ce94ad54f9791ac7b0ebf6f4036bff6c79), 
	
	
	
	
	x
	
	2
	
	
	+
	
	y
	
	2
	
	
	+
	2
	
	z
	
	2
	
	
	=
	16
	
	
	{\textstyle x^{2}+y^{2}+2z^{2}=16}
	
	![{\textstyle x^{2}+y^{2}+2z^{2}=16}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84686b9dcda37f369eea56991236263c2a9de909);
	3. u
	=
	
	∑
	
	i
	=
	1
	
	
	k
	
	
	
	a
	
	i
	
	
	
	x
	
	i
	
	
	2
	
	
	
	
	{\textstyle u=\sum \limits \_{i=1}^{k}a\_{i}x\_{i}^{2}}
	
	![{\textstyle u=\sum \limits _{i=1}^{k}a_{i}x_{i}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3b99d12ee0032f9ceed49b13a60ce96318791f), 
	
	
	
	
	∑
	
	i
	=
	1
	
	
	k
	
	
	
	x
	
	i
	
	
	=
	1
	
	
	{\textstyle \sum \limits \_{i=1}^{k}x\_{i}=1}
	
	![{\textstyle \sum \limits _{i=1}^{k}x_{i}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb5ca3b9867373e2fafdbb2b03a4566e3dfdd01), 
	
	
	
	
	a
	
	i
	
	
	>
	0
	
	
	{\textstyle a\_{i}>0}
	
	![{\textstyle a_{i}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38bcfd15c6b9a443175f08979d756ae989ab5b2d);


#### Section 3


1. Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: 




∬

D


f
(
x
;
y
)

d
x

d
y


{\textstyle \iint \limits \_{D}f(x;y)\,dx\,dy}

![{\textstyle \iint \limits _{D}f(x;y)\,dx\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b52bb56df4ec7f5416ec2e32f0903b2e1197b9) where 



D
=

{

(
x
;
y
)

|


x

2


+

y

2


≤
9
,


x

2


+
(
y
+
4

)

2


≥
25




}



{\textstyle D=\left\{(x;y)\left|x^{2}+y^{2}\leq 9,\,x^{2}+(y+4)^{2}\geq 25\right.\right\}}

![{\textstyle D=\left\{(x;y)\left|x^{2}+y^{2}\leq 9,\,x^{2}+(y+4)^{2}\geq 25\right.\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf78bafb60b17a34676ba32d3cda4bb45f3c75e).
2. Represent integral 



I
=


∭

D


f
(
x
;
y
;
z
)

d
x

d
y

d
z



{\textstyle I=\displaystyle \iiint \limits \_{D}f(x;y;z)\,dx\,dy\,dz}

![{\textstyle I=\displaystyle \iiint \limits _{D}f(x;y;z)\,dx\,dy\,dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/860cffcd0fc93fae6ac55ac594c5fa8928880ffb) as iterated integrals with all possible (i.e. 6) orders of integration; 



D


{\textstyle D}

![{\textstyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e5200f518cb5afe304ec42ffdd4f6c63c702f77) is bounded by 



x
=
0


{\textstyle x=0}

![{\textstyle x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95946e5db8fcf9ae99523bcd4f83433c8b2e441e), 



x
=
a


{\textstyle x=a}

![{\textstyle x=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70e0fe729bde9223f1b509db0471471c21261b82), 



y
=
0


{\textstyle y=0}

![{\textstyle y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4b2729074b444ef801a0afa24160d074b52eec), 



y
=


a
x




{\textstyle y={\sqrt {ax}}}

![{\textstyle y={\sqrt {ax}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acdf89c1a5d352bac27ef5bd553c3c3e2b07f8e3), 



z
=
0


{\textstyle z=0}

![{\textstyle z=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0829ff59a6fdc19b44396956e8767fac4ba87ba3), 



z
=
x
+
y


{\textstyle z=x+y}

![{\textstyle z=x+y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a04ad39377ab70df2722495d428ad59389561311).
3. Change order of integration in the iterated integral 




∫

0



2



d
y

∫

y



4
−

y

2





f
(
x
;
y
)

d
x


{\textstyle \int \limits \_{0}^{\sqrt {2}}dy\int \limits \_{y}^{\sqrt {4-y^{2}}}f(x;y)\,dx}

![{\textstyle \int \limits _{0}^{\sqrt {2}}dy\int \limits _{y}^{\sqrt {4-y^{2}}}f(x;y)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed028a999470345e5f3d630c5d992c8c67156e84).
4. Find the volume of a solid given by 



0
≤
z
≤

x

2




{\textstyle 0\leq z\leq x^{2}}

![{\textstyle 0\leq z\leq x^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/980fd1ccb56e85661acedd630abbd390e4b2003a), 



x
+
y
≤
5


{\textstyle x+y\leq 5}

![{\textstyle x+y\leq 5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6171ff088c858f3829d9dd5b3139d11be00a5d7f), 



x
−
2
y
≥
2


{\textstyle x-2y\geq 2}

![{\textstyle x-2y\geq 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1f70dd26698483c4066479e1bb099241cb9f1c), 



y
≥
0


{\textstyle y\geq 0}

![{\textstyle y\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ab86d638e82193271bc685d8d977cbbcf4e65a).


#### Section 4


1. Find line integrals of a scalar fields 





∫

Γ


(
x
+
y
)

d
s



{\textstyle \displaystyle \int \limits \_{\Gamma }(x+y)\,ds}

![{\textstyle \displaystyle \int \limits _{\Gamma }(x+y)\,ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7a626ecc11638cd879e8c5dd84deec5f4ec95e) where 



Γ


{\textstyle \Gamma }

![{\textstyle \Gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f677684faf84745068a3b602896ca59b0766be4a) is boundary of a triangle with vertices 



(
0
;
0
)


{\textstyle (0;0)}

![{\textstyle (0;0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cabde45f1dd10b936518c3de98d232d7c46150), 



(
1
;
0
)


{\textstyle (1;0)}

![{\textstyle (1;0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97e7e03a49a76bc809417791254eb6c78fcc0836) and 



(
0
;
1
)


{\textstyle (0;1)}

![{\textstyle (0;1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/047498b8de986e906f31c93d8870f2ad77819d71).
2. Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at 



A


{\textstyle A}

![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31) and finishes at 



B


{\textstyle B}

![{\textstyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de0b47ffc21636dc2df68f6c793177a268f10e9b): 





∫

Γ



(


x

4


+
4
x

y

3



)


d
x
+

(

6

x

2



y

2


−
5

y

4



)


d
y



{\textstyle \displaystyle \int \limits \_{\Gamma }\left(x^{4}+4xy^{3}\right)\,dx+\left(6x^{2}y^{2}-5y^{4}\right)\,dy}

![{\textstyle \displaystyle \int \limits _{\Gamma }\left(x^{4}+4xy^{3}\right)\,dx+\left(6x^{2}y^{2}-5y^{4}\right)\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9479ff2e89c48d84cd99758176960aefefa3c2a3), 



A
(
−
2
;
−
1
)


{\textstyle A(-2;-1)}

![{\textstyle A(-2;-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28816a18116ec2bcd546b276cb0ac6201873d963), 



B
(
0
;
3
)


{\textstyle B(0;3)}

![{\textstyle B(0;3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67293f2836c5adc3c391fa564fcdc2fac94a2562);


### Final assessment


#### Section 1


1. Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. 




∑

n
=
1


∞



e

−
n

(


x

2


+
2
sin
⁡
x

)





{\textstyle \sum \limits \_{n=1}^{\infty }e^{-n\left(x^{2}+2\sin x\right)}}

![{\textstyle \sum \limits _{n=1}^{\infty }e^{-n\left(x^{2}+2\sin x\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5683d5e640c22a7cd5b1d3520409c431056e30b0), 




Δ

1


=
(
0
;
1
]


{\textstyle \Delta \_{1}=(0;1]}

![{\textstyle \Delta _{1}=(0;1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b80239da7d74486d76788367da82b612e233dcf6), 




Δ

2


=
[
1
;
+
∞
)


{\textstyle \Delta \_{2}=[1;+\infty )}

![{\textstyle \Delta _{2}=[1;+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/767b97f867e082dfede53759fa2df79cd53a3869);
2. Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. 




∑

n
=
1


∞





n

x

3





x

2


+

n

2







{\textstyle \sum \limits \_{n=1}^{\infty }{\frac {\sqrt {nx^{3}}}{x^{2}+n^{2}}}}

![{\textstyle \sum \limits _{n=1}^{\infty }{\frac {\sqrt {nx^{3}}}{x^{2}+n^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/695d67fad209cd46deb8627d55eb112673a97d93), 




Δ

1


=
(
0
;
1
)


{\textstyle \Delta \_{1}=(0;1)}

![{\textstyle \Delta _{1}=(0;1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c54e8b5a3c4d79037b0711372b6689de58b6e4d), 




Δ

2


=
(
1
;
+
∞
)


{\textstyle \Delta \_{2}=(1;+\infty )}

![{\textstyle \Delta _{2}=(1;+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3da88d3477a5cc6c7f3a8fcfd468e4202bf6d1b9)


#### Section 2


1. Find all points where the differential of a function 



f
(
x
;
y
)
=
(
5
x
+
7
y
−
25
)

e

−

x

2


−
x
y
−

y

2






{\textstyle f(x;y)=(5x+7y-25)e^{-x^{2}-xy-y^{2}}}

![{\textstyle f(x;y)=(5x+7y-25)e^{-x^{2}-xy-y^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcef5187dd210178de071072338fc887676cb0c6) is equal to zero.
2. Show that function 



φ
=
f

(



x
y


;

x

2


+
y
−

z

2



)



{\textstyle \varphi =f\left({\frac {x}{y}};x^{2}+y-z^{2}\right)}

![{\textstyle \varphi =f\left({\frac {x}{y}};x^{2}+y-z^{2}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb557bb21657f2b0c8229adce9ee9804acd1a18) satisfies the equation 



2
x
z

φ

x


+
2
y
z

φ

y


+

(

2

x

2


+
y

)


φ

z


=
0


{\textstyle 2xz\varphi \_{x}+2yz\varphi \_{y}+\left(2x^{2}+y\right)\varphi \_{z}=0}

![{\textstyle 2xz\varphi _{x}+2yz\varphi _{y}+\left(2x^{2}+y\right)\varphi _{z}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3521dcad724a9685111a47bef3a86ae868278f4).
3. Find maxima and minima of function 



u
=
2

x

2


+
12
x
y
+

y

2




{\textstyle u=2x^{2}+12xy+y^{2}}

![{\textstyle u=2x^{2}+12xy+y^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09a2dd7f0d957f03692af28142af427d5e1791df) under condition that 




x

2


+
4

y

2


=
25


{\textstyle x^{2}+4y^{2}=25}

![{\textstyle x^{2}+4y^{2}=25}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2161246eedccf6d799aa1e5734b118c315f7150). Find the maximum and minimum value of a function
4. u
=

(


y

2


−

x

2



)


e

1
−

x

2


+

y

2






{\textstyle u=\left(y^{2}-x^{2}\right)e^{1-x^{2}+y^{2}}}

![{\textstyle u=\left(y^{2}-x^{2}\right)e^{1-x^{2}+y^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/932e00eb2464b47b893d6fb18846cfcccb4e6d6b) on a domain given by inequality 




x

2


+

y

2


≤
4


{\textstyle x^{2}+y^{2}\leq 4}

![{\textstyle x^{2}+y^{2}\leq 4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90eb8b71c5ce09a4ec482966f3de5479978b1e3d);


#### Section 3


1. Domain 



G


{\textstyle G}

![{\textstyle G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/febc2b9ff73bcca7b3fdb1432fddd1cdf3c8403c) is bounded by lines 



y
=
2
x


{\textstyle y=2x}

![{\textstyle y=2x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d06a41ec951090ac63c9e967e4d2d827376c9ca), 



y
=
x


{\textstyle y=x}

![{\textstyle y=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93343635ea42bd77e3e1ce347195e23ade09b5d3) and 



y
=
2


{\textstyle y=2}

![{\textstyle y=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/887628e811c56cc94aaccb0f5e4c773f19231cf9). Rewrite integral 




∬

G


f
(
x
)

d
x

d
y


{\textstyle \iint \limits \_{G}f(x)\,dx\,dy}

![{\textstyle \iint \limits _{G}f(x)\,dx\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc886f1ea76c3bc7f81d3c334f7620147794855) as a single integral.
2. Represent the integral 





∬

G


f
(
x
;
y
)

d
x

d
y



{\textstyle \displaystyle \iint \limits \_{G}f(x;y)\,dx\,dy}

![{\textstyle \displaystyle \iint \limits _{G}f(x;y)\,dx\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11428df0d608e522a446f7cf8ee68cdbdbdee29) as iterated integrals with different order of integration in polar coordinates if 



G
=

{

(
x
;
y
)

|


a

2


≤

x

2


+

y

2


≤
4

a

2


;


|

x

|

−
y
≥
0




}



{\textstyle G=\left\{(x;y)\left|a^{2}\leq x^{2}+y^{2}\leq 4a^{2};\,|x|-y\geq 0\right.\right\}}

![{\textstyle G=\left\{(x;y)\left|a^{2}\leq x^{2}+y^{2}\leq 4a^{2};\,|x|-y\geq 0\right.\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34dc3c913392b2188d0f55ce7885d44baa8a8f59).
3. Find the integral making an appropriate substitution: 





∭

G



(


x

2


−

y

2



)


(

z
+

x

2


−

y

2



)


d
x

d
y

d
z



{\textstyle \displaystyle \iiint \limits \_{G}\left(x^{2}-y^{2}\right)\left(z+x^{2}-y^{2}\right)\,dx\,dy\,dz}

![{\textstyle \displaystyle \iiint \limits _{G}\left(x^{2}-y^{2}\right)\left(z+x^{2}-y^{2}\right)\,dx\,dy\,dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7f604b225ae19524d76d6342cb44f0c7fe07ff), 



G
=

{

(
x
;
y
;
z
)

|

x
−
1
<
y
<
x
;

1
−
x
<
y
<
2
−
x
;

1
−

x

2


+

y

2


<
z
<

y

2


−

x

2


+
2
x




}



{\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\}}

![{\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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28659bdf51117253f1eea6a8c57f76540a6ec159).


#### Section 4


1. Find line integrals of a scalar fields 





∫

Γ


(
x
+
y
)

d
s



{\textstyle \displaystyle \int \limits \_{\Gamma }(x+y)\,ds}

![{\textstyle \displaystyle \int \limits _{\Gamma }(x+y)\,ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7a626ecc11638cd879e8c5dd84deec5f4ec95e) where 



Γ


{\textstyle \Gamma }

![{\textstyle \Gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f677684faf84745068a3b602896ca59b0766be4a) is boundary of a triangle with vertices 



(
0
;
0
)


{\textstyle (0;0)}

![{\textstyle (0;0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cabde45f1dd10b936518c3de98d232d7c46150), 



(
1
;
0
)


{\textstyle (1;0)}

![{\textstyle (1;0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97e7e03a49a76bc809417791254eb6c78fcc0836) and 



(
0
;
1
)


{\textstyle (0;1)}

![{\textstyle (0;1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/047498b8de986e906f31c93d8870f2ad77819d71).
2. Use divergence theorem to find the following integrals 





∬

S


(
1
+
2
x
)

d
y

d
z
+
(
2
x
+
3
y
)

d
z

d
x
+
(
3
y
+
4
z
)

d
x

d
y



{\textstyle \displaystyle \iint \limits \_{S}(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy}

![{\textstyle \displaystyle \iint \limits _{S}(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a9a7b00f2cc15f508b9bf55224d55b9d1e7e14) where 



S


{\textstyle S}

![{\textstyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e10a3c52d186162ec8910ebc0288ce982aef842f) is the outer surface of a tetrahedron 





x
a


+


y
b


+


z
c


≤
1


{\textstyle {\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}\leq 1}

![{\textstyle {\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1377463cd4c0d1b190d99150ad4c7cecd93ca22f), 



x
≥
0


{\textstyle x\geq 0}

![{\textstyle x\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c434fc1e2ab777786469de853c75e616007b3eb4), 



y
≥
0


{\textstyle y\geq 0}

![{\textstyle y\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ab86d638e82193271bc685d8d977cbbcf4e65a), 



z
≥
0


{\textstyle z\geq 0}

![{\textstyle z\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7298578ca3a51011394eff000d2c36a4b6cf9c7b);


### The retake exam


Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.