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BSc: Mathematical Analysis II.s23
Contents
- 1 Mathematical Analysis II
- 1.1 Short Description
- 1.2 Course Topics
- 1.3 Intended Learning Outcomes (ILOs)
- 1.4 Grading
- 1.5 Resources, literature and reference materials
- 1.6 Activities and Teaching Methods
- 1.7 Formative Assessment and Course Activities
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. Greenβs Theorem 4. Circulation and Stokeβs 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
- 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
- Derive the Maclaurin expansion for
f ( x )
1 +
e
β 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)
o
(
x
3
)
{\textstyle o\left(x^{3}\right)}
lim
x β 0
lim
y β 0
u ( x ; y )
{\textstyle \lim \limits _{x\to 0}\lim \limits _{y\to 0}u(x;y)}
lim
y β 0
lim
x β 0
u ( x ; y )
{\textstyle \lim \limits _{y\to 0}\lim \limits _{x\to 0}u(x;y)}
lim
( x ; y ) β ( 0 ; 0 )
u ( x ; y )
{\textstyle \lim \limits _{(x;y)\to (0;0)}u(x;y)}
u ( x ; y )
x
2
y + x
y
2
x
2
β x y +
y
2
{\textstyle u(x;y)={\frac {x^{2}y+xy^{2}}{x^{2}-xy+y^{2}}}}
Section 2
- Find the differential of a function: (a)
u ( x ; y )
ln β‘
(
x +
x
2
y
2
)
{\textstyle u(x;y)=\ln \left(x+{\sqrt {x^{2}+y^{2}}}\right)}
u ( x ; y )
ln β‘ sin β‘
x + 1
y
{\textstyle u(x;y)=\ln \sin {\frac {x+1}{\sqrt {y}}}}
u ( x ; y )
{\textstyle u(x;y)}
x
3
- 2
y
3
u
3
β 3 x y u + 2 y β 3
0
{\textstyle x^{3}+2y^{3}+u^{3}-3xyu+2y-3=0}
M ( 1 ; 1 ; 1 )
{\textstyle M(1;1;1)}
N ( 1 ; 1 ; β 2 )
{\textstyle N(1;1;-2)}
x
2
y
3
z
4
{\textstyle u=x^{2}y^{3}z^{4}}
,
2
x
+
3
y
+
4
z
=
18
{\textstyle 2x+3y+4z=18}
,
x
>
0
{\textstyle x>0}
,
y
>
0
{\textstyle y>0}
,
z
>
0
{\textstyle z>0}
;
2. u
=
x
β
y
+
2
z
{\textstyle u=x-y+2z}
,
x
2
+
y
2
+
2
z
2
=
16
{\textstyle x^{2}+y^{2}+2z^{2}=16}
;
3. u
=
β
i
=
1
k
a
i
x
i
2
{\textstyle u=\sum \limits \_{i=1}^{k}a\_{i}x\_{i}^{2}}
,
β
i
=
1
k
x
i
=
1
{\textstyle \sum \limits \_{i=1}^{k}x\_{i}=1}
,
a
i
>
0
{\textstyle a\_{i}>0}
;
Section 3
- Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration:
β¬
D
f ( x ; y )
d x
d y
{\textstyle \iint \limits _{D}f(x;y),dx,dy}
D
{
( x ; y )
|
x
2
y
2
β€ 9 ,
x
2
- ( y
- 4
)
2
β₯ 25
}
{\textstyle D=\left{(x;y)\left|x^{2}+y^{2}\leq 9,,x^{2}+(y+4)^{2}\geq 25\right.\right}}
I
β
D
f ( x ; y ; z )
d x
d y
d z
{\textstyle I=\displaystyle \iiint \limits _{D}f(x;y;z),dx,dy,dz}
D
{\textstyle D}
x
0
{\textstyle x=0}
x
a
{\textstyle x=a}
y
0
{\textstyle y=0}
y
a x
{\textstyle y={\sqrt {ax}}}
z
0
{\textstyle z=0}
z
x + y
{\textstyle z=x+y}
β«
0
2
d y
β«
y
4 β
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}
0 β€ z β€
x
2
{\textstyle 0\leq z\leq x^{2}}
x + y β€ 5
{\textstyle x+y\leq 5}
x β 2 y β₯ 2
{\textstyle x-2y\geq 2}
y β₯ 0
{\textstyle y\geq 0}
Section 4
- Find line integrals of a scalar fields
β«
Ξ
( x + y )
d s
{\textstyle \displaystyle \int \limits _{\Gamma }(x+y),ds}
Ξ
{\textstyle \Gamma }
( 0 ; 0 )
{\textstyle (0;0)}
( 1 ; 0 )
{\textstyle (1;0)}
( 0 ; 1 )
{\textstyle (0;1)}
A
{\textstyle A}
B
{\textstyle B}
β«
Ξ
(
x
4
- 4 x
y
3
)
d x +
(
6
x
2
y
2
β 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}
A ( β 2 ; β 1 )
{\textstyle A(-2;-1)}
B ( 0 ; 3 )
{\textstyle B(0;3)}
Final assessment
Section 1
- Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer.
β
n
1
β
e
β n
(
x
2
- 2 sin β‘ x
)
{\textstyle \sum \limits _{n=1}^{\infty }e^{-n\left(x^{2}+2\sin x\right)}}
Ξ
1
= ( 0 ; 1 ]
{\textstyle \Delta _{1}=(0;1]}
![{\textstyle \Delta _{1}=(0;1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b80239da7d74486d76788367da82b612e233dcf6),
Ξ
2
= [ 1 ; + β )
{\textstyle \Delta _{2}=[1;+\infty )}
Ξ
1
= ( 0 ; 1 )
{\textstyle \Delta _{1}=(0;1)}
Ξ
2
= ( 1 ; + β )
{\textstyle \Delta _{2}=(1;+\infty )}
Section 2
- Find all points where the differential of a function
f ( x ; y )
( 5 x + 7 y β 25 )
e
β
x
2
β x y β
y
2
{\textstyle f(x;y)=(5x+7y-25)e^{-x^{2}-xy-y^{2}}}
Ο
f
(
x y
;
x
2
- y β
z
2
)
{\textstyle \varphi =f\left({\frac {x}{y}};x^{2}+y-z^{2}\right)}
2 x z
Ο
x
- 2 y z
Ο
y
(
2
x
2
- y
)
Ο
z
= 0
{\textstyle 2xz\varphi _{x}+2yz\varphi _{y}+\left(2x^{2}+y\right)\varphi _{z}=0}
u
2
x
2
- 12 x y
y
2
{\textstyle u=2x^{2}+12xy+y^{2}}
x
2
- 4
y
2
= 25
{\textstyle x^{2}+4y^{2}=25}
. Find the maximum and minimum value of a function
4. u
(
y
2
β
x
2
)
e
1 β
x
2
y
2
{\textstyle u=\left(y^{2}-x^{2}\right)e^{1-x^{2}+y^{2}}}
x
2
y
2
β€ 4
{\textstyle x^{2}+y^{2}\leq 4}
Section 3
- Domain
G
{\textstyle G}
y
2 x
{\textstyle y=2x}
y
x
{\textstyle y=x}
y
2
{\textstyle y=2}
β¬
G
f ( x )
d x
d y
{\textstyle \iint \limits _{G}f(x),dx,dy}
β¬
G
f ( x ; y )
d x
d y
{\textstyle \displaystyle \iint \limits _{G}f(x;y),dx,dy}
G
{
( x ; y )
|
a
2
β€
x
2
y
2
β€ 4
a
2
;
|
x
|
β y β₯ 0
}
{\textstyle G=\left{(x;y)\left|a^{2}\leq x^{2}+y^{2}\leq 4a^{2};,|x|-y\geq 0\right.\right}}
β
G
(
x
2
β
y
2
)
(
z +
x
2
β
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}
G
{
( x ; y ; z )
|
x β 1 < y < x ;
1 β x < y < 2 β x ;
1 β
x
2
y
2
< z <
y
2
β
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}}
Section 4
- Find line integrals of a scalar fields
β«
Ξ
( x + y )
d s
{\textstyle \displaystyle \int \limits _{\Gamma }(x+y),ds}
Ξ
{\textstyle \Gamma }
( 0 ; 0 )
{\textstyle (0;0)}
( 1 ; 0 )
{\textstyle (1;0)}
( 0 ; 1 )
{\textstyle (0;1)}
β¬
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}
S
{\textstyle S}
x a
y b
z c
β€ 1
{\textstyle {\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}\leq 1}
x β₯ 0
{\textstyle x\geq 0}
y β₯ 0
{\textstyle y\geq 0}
z β₯ 0
{\textstyle z\geq 0}
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.