Study Program Name Bachelor, mathematics Department, FMKSD-ITS Course Name Complex Variable
Course Code KM184602
Semester 6
SKS 3
Supporting Lecturer Drs. Sentot Didik Surjanto, M.Si
Materials • Complex numbers
• Complex function
Learning Outcome
[C2]
Able to explain basic concepts of mathematics that includes the concept of a proof construction both logically and analytically, modeling and solving the simple problems, as well as the basic of computing.
[C3]
Able to solve problems based on theoretical concepts in at least one field of mathematics: analysis and algebra, modeling and system optimization, and computing science.
[C4]
Able to illustrate the framework of mathematical thinking in particular areas such as analysis, algebra, modeling, system optimization and computing science to solve real problems, mainly in the areas of environment, marine, energy and information technology.
COURSE LEARNING OUTCOME
1. Students are able to explain the nature of algebra in complex numbers, determine limits, continuity and derivation of complex functions and can explain the properties of elementary functions: exponential functions, logarithms, and trigonometry, hyperbolic functions, and trigonometric invers
2. Students are able to calculate the integral complex functions using appropriate properties and theorems
3. Students are able to explain the mapping / transformation by elementary functions and conformal mapping / transformation
4. Students are able to explain the residual theorem and its use to compute the integral complex functions
5. Students are able to investigate series convergence, decompose complex functions in power series, Taylor, Maclaurin and Lourent series
Meets Sub Course Learning
Outcome Breadth of Materials Learning Methods Time
Estimation Student Learning Experiences
Assessment Criteria
and Indicator Weight ing Assess
ment (%) 1,2 Students are able to
explain the notion of complex numbers, the nature of algebra and its operations.
Lecture Contract
o Understanding complex number systems
o Notation of complex number forms
o Operation of complex numbers
o Basic concepts in topology in complex fields
Ref: {1} Chapter 1 and Ref: {1} Chapter 1
o Lecture
o Group discussion o Students work in front of the discussion result class
2x(3x50’)] o Resume of course material o TaSks concerning complex numbers
o Able to solve problems about complex numbers o Understanding the notation of complex number forms with algebraic operations
10%
3 Students are able to explain the meaning of function, limit, continuous, derivative of complex functions.
Lecture Contract
o Understanding complex functions
o Limit and continuous complex functions
o The derivation of complex functions
Ref: {1} Chapter 2 and Ref:
{2} Chapter 2
o Lecture
o Group discussion o Students work in front of the discussion result class
1x(3x50’)] o Resume of course material o TaSks about complex functions
o Able to solve problems about function, limit, and continuous.
o Understand the function derivative.
15%
4 Students are able to explain the meaning of Cauchy_Riemann equation, analytic function, harmonic.
Lecture Contract o Understanding the Cauchy_Riemann equation o Understanding of analytic functions
o Understanding the harmonic function
Ref: {1} Chapter 3 and Ref:
{2} Chapter 3
o Lecture
o Group discussion o Students work in front of the discussion result class
1x(3x50’)] o Resume of course material o Duties on derivatives, analytic functions, harmonics
o Able to solve problems about the derivative
o Understand the analytic function, harmonics
10%
5, 6 Students are able to explain elementary functions: exponential, logarithmic,
trigonometric, inverse trigonometric,
hyperbolic and complex ranks
Lecture Contract o Understanding elementary functions o Understanding exponential functions, logarithms, trigonometry o Understanding
trigonometric, hyperbolic and complex power inverse functions
Ref: {1} Chapter 4 and Ref: {2} Chapter 4
o Lecture
o Group discussion o Students work in front of the discussion result class
2x(3x50’)] o Resume of course material o TaSks about elemental functions o Evaluate chapters 1 and 2
Able to solve problems about elementary functions
15%
7 Students are able to explain the integral complex functions: path integrals, Cauchy-Goursat theorem, Cauchy integral, Morera's theorem, Liouville's theorem
Lecture Contract
o Understanding the integral of complex functions
o Understanding the integrals of the trajectory
o Understanding of Cauchy-Goursat's theorem, Cauchy integral, Morera's theorem, Liouville's theorem.
Ref: {1} Chapter 5 and Ref:
{2} Chapter 5
o Lecture
o Group discussion o Students work in front of the discussion result class
1x(3x50’)] o Resume of course material o The task of integral complex functions o Discuss evaluation chapters 1 and 2
Be able to solve problems about integral complex functions
15
8 MIDTERM EXAM
Reference Main :
1. Churchil, R., ”Complex Variables and Applications 8th edition”, McGraw-Hill, New York, 2009.
9, 10 Students are able to explain the power series: Taylor series, Maclaurin, Laurent, series operations
Lecture Contract
o Understanding the power series
o Understanding Taylor series, Maclaurin
o Understanding Laurent, multiplication operations and the division of the series Ref: {1} Chapter 6 and Ref:
{2} Chapter 6
o Lecture
o Group discussion o Students work in front of the discussion result class
2x(3x50’)] o Resume of course material o The task of power series
o Able to solve questions about power series
10%
11,
12, 13 Students are able to explain poles and residues: use of residues in integral.
Lecture Contract
o Understanding poles and residues
o Understanding the use of residues in integral
o Integral notions of per unit circle, pagef disc and unnatural
Ref: {1} Chapter 7 and Ref:
{2} Chapter 7
o Lecture
o Group discussion o Students work in front of the discussion result class
3x(3x50’)] o Resume of course material o The task of using residues in integral
Be able to solve integral problems with residues
15%
14, 15 Students are able to explain the
transformation of elinsnter and conformal functions.
Lecture Contract o Understanding the transformation of elementary functions o Understanding conformal transformation
Ref: {1} Chapter 7,8 and Ref: {2} Chapter 7.8
o Lecture
o Group discussion o Students work in front of the discussion result class
2x(3x50’)] o Resume of course material o TaSks regarding the transformation of elinsnter and conformal functions
Able to solve problems of functional and conformal functional transformation
15%
16 Final Exam 10
2. Mathews, J.H, “Complex Variables for Mathematics and Engineering”, 6th edition, WM C Brown Publiser, Iowa, 2010.
Supporting :
1. Poliouras, J.D., Meadows D. S, ”Complex Variables for Scientists and Engineers 2nd edition ”, New York, 2014.
Course
Course Name : Complex Variable
Course Code : KM184602
Credit : 3
Semester : 6
Description of Course
The subjects of the complex function variables address the problem: complex numbers, complex mapping, limiting, continuous, derivative, complex integral, Green Theorem, Cauchy, Morera and Liouvile, convergence / divergence sequences and series, singularities, residual theorems and their use in complex integrals, conformal mapping.
Learning Outcome
[C2]
Able to explain basic concepts of mathematics that includes the concept of a proof construction both logically and analytically, modeling and solving the simple problems, as well as the basic of computing.
[C3]
Able to solve problems based on theoretical concepts in at least one field of mathematics: analysis and algebra, modeling and system optimization, and computing science.
[C4]
Able to illustrate the framework of mathematical thinking in particular areas such as analysis, algebra, modeling, system optimization and computing science to solve real problems, mainly in the areas of environment, marine, energy and information technology.
Course Learning Outcome
1. Students are able to explain the nature of algebra in complex numbers, determine limits, continuity and derivation of complex functions and can explain the properties of elementary functions: exponential functions, logarithms, and trigonometry, hyperbolic functions, and trigonometric invers
2. Students are able to calculate the integral complex functions using appropriate properties and theorems
3. Students are able to explain the mapping / transformation by elementary functions and conformal mapping / transformation
4. Students are able to explain the residual theorem and its use to compute the integral complex functions
5. Students are able to investigate series convergence, decompose complex functions in power series, Taylor, Maclaurin and Lourent series
Main Subject
Complex number system, complex variable function, limit, continuity, derivative, analytic function and harmonic function, elementary functions:
exponential, logarithm, trigonometry, hyperbolic, and trigonometric inverse, complex integration, contour, theorem: Green, Cauchy, Morera and Liouvile, convergence / divergence sequence and series, singularity, residual theorem and its use in complex function integral, conformal mapping
Prerequisites
Analysis I Reference
1. Churchil, R., ”Complex Variables and Applications 8th edition”, McGraw-Hill, New York, 2009.
2. Mathews, J.H, “Complex Variables for Mathematics and Engineering”, 6th edition, WM C Brown Publiser, Iowa, 2010.
Supporting Reference
1. Poliouras, J.D., Meadows D. S, ”Complex Variables for Scientists and Engineers 2nd edition ”, New York, 2014.
Study Program Name Bachelor, Mathematics Department,FMKSD-ITS Course Name Introduction to Dynamic Optimization
Course Code KM184716
Semester 7
Sks 2
Supporting Lecturer Dr. Dra. Mardlijah,MT
Materials Calculus of Variatons
Optimal Control
Learning Outcome
[C3] Able to solve problems based on theoretical concepts in at least one field of mathematics: analysis and algebra, modeling and system optimization, and computing science.
[C4]
Able to illustrate the framework of mathematical thinking in particular areas such as analysis, algebra, modeling, system optimization and computing science to solve real problems, mainly in the areas of environment, marine, energy and information technology.
[C5] Able to choose decisions and alternative solutions using data and information analysis based on an attitude of leadership, creativity and have high integrity in completing work individually or in a team.
COURSE LEARNING OUTCOME
.
1. Students are able to follow the development and apply Mathematics and able to communicate actively and correctly either oral or written.
2. Students are able to explain basic and advanced principles of the Theory they understand especially in relation to the optimization design formulation and the method of completion
3. Students are able to explain intelligently and creatively about the significant role of the optimization system in the field of related knowledge clusters or other fields.
Meets Sub Course Learning
Outcome Breadth of Materials Learning Methods Time
Estimation Student Learning Experiences
Assessment Criteria
and Indicator Weighting Assessment
(%) 1 Students are able to
model and categorize static and dynamic optimization problems.
Lecture Contract Static and Dynamic optimization
[1]: Subchan Chapter II
Propaedeutics ,
simple case study 1x(2x50”) Writing about the solution of some given problems
Good ability to explain differences in static and dynamic optimization problems and apply them
10 %
2,3 Student are ablet to differentiate simple function and functional problems
Function and Functional differences
[1]: Naidu Chapter II [2]: Krasnov Chapter II
‐ Lectures
‐ Exercises/Review
2x(2x50”) Writing about the solution of some given problems
Able to understand function and functional differences
15%
4 Students are able to explain the concepts of optimal function and functional
Optimal function and functional
[1]: Naidu Chapter II.2
‐ Lectures
‐ Exercise
1x(2x50”) Writing about the solution of some given problems
Good ability to explain the concept of optimal function and functional
5 %
5-6-7 Students are able to explain the basics of variational and classify the real problems into Euler-Lagrange cases
Know time and state Decrease in Euler-Lagrange The Euler-Lagrange Cases [1]: Naidu Chapter II.3
‐ Lectures
‐ Exercise
3x(2x50”) Writing about the solution of some given problems
Good ability to explain basic of variational and decrease of Euler-Lagrange
20%
MIDTERM EXAM
9,10
Students are able to explain and evaluate optimal function and functional with constraints
Optimal function and functional with constraints [1]: Naidu Chapter II.5-II.6
‐ Lectures,
‐ Review,
‐ Practice
2x(2x50”)
Writing about the solution of some given problems
Good ability to explain the difference between array-based and linked stack implementation
10%
11-12-13
Students are able to apply variational approach to optimal control and evaluate it
Cariational approach to optimal control
[1]: Naidu Chapter II.7-II.8
‐ Lectures
‐ Review,
‐ Practice
3x(2x50”) ‐ Source code of practice result
‐ Writing about the solution of
Good ability to apply a variational approach to optimal control and to evaluate
15%
Reference Main :
1. Naidu, D.S, Optimal Control Systems, CRC Press, 2002
2. Bolza, O. Lectures on the Calculus of Variations, American Mathematical Society; 3 edition (October 31, 2000)
Supporting :
1. Subchan, S and Zbikowski, R., Computational Optimal Control: Tools and Practice, Wiley, 2009.
some given problems
14-15
Students are able to explain, apply optimal control in real problem and evaluate the result
Case study Lectures
Project I 2x(2x50”) Presentation Good ability to explain, apply optimal control in real problems and evaluate results
25%
(16) Final Exam
STUDENT LEARNING EVALUATION PLAN
Course : Introduction to Dynamic Optimization, Code: KM184716, sks:2 sks, smt:7 Learning outcome :
1. Able to identify simple problems, form mathematical models and solve them.
2. Able to identify problems, form mathematical models and solve them.
3. Able to analyze the system and optimize its performance
4. Able to understand mathematical problems, analyze and solve them.
5. Able to analyze a phenomenon through a mathematical model and solve it
6. Able to apply mathematical thinking to solve optimization problems both analytically and empirically.
7. Able to observe, recognize, formulate and solve problems through mathematical approaches
8. Able to analyze structurally a system / problem, reconstruct, and modify it into a mathematical model;
Me ets
Specific Learning Objective (Sub‐Competence)
Elements of Competency in Assessment Number of Questio
ns
Form of
Assessment %
Cognitive Psychomotor Affective
C1 C2 C3 C4 C5 C6 C7 P1 P2 P3 P4 P5 A1 A2 A3 A4 A5 1 Students are able to model categorizing static
and dynamic optimization problems. v v v Write about the
solutions to some of the problems given
10%
2,3 Students are able to model categorizing static
and dynamic optimization problems. v v v Write about the
solutions to some 15%
of the problems given
4 Students are able to explain the concept of
optimal function and function v v v Write about the
solutions to some of the problems given
5%
5,6, 7
- Students are able to explain variational basics and classify real problems into Euler-Lagrange cases
v v v Write about the
solutions to some of the problems given
20%
8 Mid Semester Evaluation
9‐
10
Students are able to explain and evaluate optimal functions and functional constraints
v v v - Writing about
solutions to several problems given
10%
11, 12, 13
Students are able to explain and evaluate optimal functions and functional constraints
v v v - Writing about
solutions to several problems given
15%
14,1 5
- Students are able to explain and apply optimal
control in real problems and evaluate the results v v v o Presentation 25%
16 Final Semester Evaluation
Number
Questions Item
Percentage 100%
Informations :
C1 : Knowledge P1 : Imitation A1 : Receiving
C2 : Comprehension P2 : Manipulation A2 : Responding
C3 : Application P3 : Precision A3 : Valuing
C4 : Analysis P4 : Articulation A4 : Organization
C5 : Syntesis & Evaluation P5 : Naturalisation A5 : Characterization C6 : Creative
Task Design Format Course : Introduction to Dynamical Optimization Semester : VII
Code : KM184716 sks : 2
Weeks : 4
1. Purpose of Task :
Students are able to explain the concept of optimal function and function 2. Task Description
a. Claim Object :
Function optimization and simple functional forms b. What to do and limitation :
B1. Determine whether the following functions are maximum or minimum
𝑓 𝑥 𝑦 2𝑥 4𝑥𝑦 2𝑦
B2. Find distance between curve 𝑦 𝑥 and 𝑦 𝑥 on interval 0,1 B3. Determine extreme conditional from 𝑓 𝑥𝑦𝑧 with condition
𝑥 𝑦 𝑧 5
𝑥𝑦 𝑦𝑧 𝑧𝑥 8
B4. Determine change in functional
𝐽 𝑦 𝑥 𝑦 𝑥 𝑦 𝑑𝑥
if given 𝑦 𝑥 𝑒 dan 𝑦 𝑥 1.
c. Method/way reference work used :
Tasks are typed in A4 paper size 12 letter spacing 1.15 normal margins.
d. Description of output of work produced/ done
Writing about the solutions to several problems given 3. Assessment criteria
No. Assessed Aspects / Concepts Score
1 Able to determine optimization (maximum and minimum values) of a function
15
2 Able to determine the distance between two functions 25 3 Able to determine extreme conditionals of functions 𝑓 𝑥𝑦𝑧
with a constraint
25
4 Able to determine changes in functional J [y (x)] with an constraint
35
Score total 100
Task Design Format Course : Introduction to Dynamical Optimization Semester : VII
Code : KM184716 sks : 2
Weeks : 12 1. Purpose of Task :
Students are able to explain and evaluate optimal functions and functional constraints 2. Description of Task
a. Claim object:
Functional
b. What to do and limitation:
B1. Determine completion from problem follows: (score 30%)
𝐽 𝑥 𝑡 𝑡𝑥 𝑥 𝑑𝑡
B2. Determine completion from the following dynamics optimization problems:
(score 35%)
min 𝐽 𝑢 𝑡 1
2 𝑥 𝑢 𝑑𝑡
with constraints
𝑥 𝑢
𝑥 0 1
B3. Determine completion from the following dynamical optimization problem:
(score 35%)
max 𝐽 𝑢 𝑡 𝑢 𝑑𝑡
with contraints
𝑥 𝑥 𝑢
𝑥 0 1
𝑥 1 0
c. Method/way the reference works is used:
Tasks are typed in A4 paper size 12 letter spacing 1.15 normal margins.
d. Description of output of work produce/done :
Write about the solutions to some of the problems given
Study Program Name Bachelor, Mathematics Department,FMKSD-ITS Course Name Introduction to Dynamic Optimization
Course Code KM184716
Semester 7
Sks 2
Supporting Lecturer Dr. Dra. Mardlijah,MT
Materials Calculus of Variatons
Optimal Control
Learning Outcome
[C3] Able to solve problems based on theoretical concepts in at least one field of mathematics: analysis and algebra, modeling and system optimization, and computing science.
[C4]
Able to illustrate the framework of mathematical thinking in particular areas such as analysis, algebra, modeling, system optimization and computing science to solve real problems, mainly in the areas of environment, marine, energy and information technology.
[C5] Able to choose decisions and alternative solutions using data and information analysis based on an attitude of leadership, creativity and have high integrity in completing work individually or in a team.
COURSE LEARNING OUTCOME
.
1. Students are able to follow the development and apply Mathematics and able to communicate actively and correctly either oral or written.
2. Students are able to explain basic and advanced principles of the Theory they understand especially in relation to the optimization design formulation and the method of completion
3. Students are able to explain intelligently and creatively about the significant role of the optimization system in the field of related knowledge clusters or other fields.
Meets Sub Course Learning
Outcome Breadth of Materials Learning Methods Time
Estimation Student Learning Experiences
Assessment Criteria
and Indicator Weighting Assessment
(%) 1 Students are able to
model and categorize static and dynamic optimization problems.
Lecture Contract Static and Dynamic optimization
[1]: Subchan Chapter II
Propaedeutics ,
simple case study 1x(2x50”) Writing about the solution of some given problems
Good ability to explain differences in static and dynamic optimization problems and apply them
10 %
2,3 Student are ablet to differentiate simple function and functional problems
Function and Functional differences
[1]: Naidu Chapter II [2]: Krasnov Chapter II
‐ Lectures
‐ Exercises/Review
2x(2x50”) Writing about the solution of some given problems
Able to understand function and functional differences
15%
4 Students are able to explain the concepts of optimal function and functional
Optimal function and functional
[1]: Naidu Chapter II.2
‐ Lectures
‐ Exercise
1x(2x50”) Writing about the solution of some given problems
Good ability to explain the concept of optimal function and functional
5 %
5-6-7 Students are able to explain the basics of variational and classify the real problems into Euler-Lagrange cases
Know time and state Decrease in Euler-Lagrange The Euler-Lagrange Cases [1]: Naidu Chapter II.3
‐ Lectures
‐ Exercise
3x(2x50”) Writing about the solution of some given problems
Good ability to explain basic of variational and decrease of Euler-Lagrange
20%
MIDTERM EXAM
9,10
Students are able to explain and evaluate optimal function and functional with constraints
Optimal function and functional with constraints [1]: Naidu Chapter II.5-II.6
‐ Lectures,
‐ Review,
‐ Practice
2x(2x50”)
Writing about the solution of some given problems
Good ability to explain the difference between array-based and linked stack implementation
10%
11-12-13
Students are able to apply variational approach to optimal control and evaluate it
Cariational approach to optimal control
[1]: Naidu Chapter II.7-II.8
‐ Lectures
‐ Review,
‐ Practice
3x(2x50”) ‐ Source code of practice result
‐ Writing about the solution of
Good ability to apply a variational approach to optimal control and to evaluate
15%
Reference Main :
1. Naidu, D.S, Optimal Control Systems, CRC Press, 2002
2. Bolza, O. Lectures on the Calculus of Variations, American Mathematical Society; 3 edition (October 31, 2000)
Supporting :
1. Subchan, S and Zbikowski, R., Computational Optimal Control: Tools and Practice, Wiley, 2009.
some given problems
14-15
Students are able to explain, apply optimal control in real problem and evaluate the result
Case study Lectures
Project I 2x(2x50”) Presentation Good ability to explain, apply optimal control in real problems and evaluate results
25%
(16) Final Exam
Course
Course Name : Introduction to Dynamic Optimization
Course Code : KM184716
Credit : 2
Semester : 7
Description of Course
The discussion of the dynamic optimization course includes the study of the basics of calculus variation, and the approach of calculus varasi on optimal control. In the learning process in the classroom learners will learn to identify problems, model. In addition to being directed to independent learning through tasks, learners are directed to cooperate in group work.
Learning Outcome
[C3] Able to solve problems based on theoretical concepts in at least one field of mathematics: analysis and algebra, modeling and system optimization, and computing science.
[C4]
Able to illustrate the framework of mathematical thinking in particular areas such as analysis, algebra, modeling, system optimization and computing science to solve real problems, mainly in the areas of environment, marine, energy and information technology.
[C5] Able to choose decisions and alternative solutions using data and information analysis based on an attitude of leadership, creativity and have high integrity in completing work individually or in a team.
Course Learning Outcome
1. Students are able to follow the development and apply Mathematics and able to communicate actively and correctly either oral or written.
2. Students are able to explain basic and advanced principles of the Theory they understand especially in relation to the optimization design formulation and the method of completion
3. Students are able to explain intelligently and creatively about the significant role of the optimization system in the field of related knowledge clusters or other fields.
Main Subject
Basic Concepts, Function and Functional, Optimum of a Function and a Functional, The Basic Variational Problem, Fixed-End Time and Fixed-End State System, Discussion on Euler-Lagrange Equation , Different Cases for Euler-Lagrange Equation, The Second Variation , Extrema of Functions with Conditions, Extrema of Functionals with Conditions, Variational Approach to Optimal Control Systems.
Prerequisites