MSc Higher Mathematics I

Table of Contents

• Lecture Information
• Requirements and the Learning Outcomes
• Grading of the Lecture
• Lecture Sources
• Content and Unit Distribution
• Lecture Table of Contents
• Related Links

Lecture Information

• The goal of this lecture is to introduce you to the tools you need to learn to tackle more advanced engineering problems.
• These could be ranging from doing circuit analysis to calculating the stress experienced by a bridge which will be covered in this lecture series as examples.

The structure for this lecture is as follows.

DESCRIPTION	VALUE
Official Name	Höhere Mathematik 1
Lecture Code	HMA
Module Code	MECH-M-1-HMA-HMA-VO
Degree	M.Sc
Program Name	Mechatronik Smart Technologies
Lecture Name	Drive Systems
Semester	1
Season	WS
Room Type	Lecture Room
Assignments	Personal Assignment Final Exam
Lecturer	Daniel T. McGuiness, Ph.D
Module Responsible	DaM
Software	Python, SageMath
Hardware	-
SWS Total	2
SWS Teaching	2
ECTS	3
Lecture Type	VO

Requirements and the Learning Outcomes

• The student should be comfortable with working with calculus and be familiar with taking derivatives and doing integration.

REQUIREMENTS	TAUGHT LECTURE	CODE	DEGREE	OUTCOME
Calculus	Mathematics I	MAT I	B.Sc	Advanced Vector Calculus
Linear Algebra	Mathematics II	MAT II	B.Sc	ODE Solving Methods
-				Understanding Transforms
-				Eigenvalues, Eigenvectors
-

Grading of the Lecture

• The lecture will have a single personal assignment comprising of a set list of questions and a final exam comprising of all the topics covered in the lecture.
• For the written exam you are allowed to write your own equation reference paper, as long as it is a single sheet of A4, double sided and contains no exercise or solutions.

ASSIGNMENT TYPE	VALUE
Personal Assignment	40
Final Exam	60
SUM	100

Lecture Sources

The following are a table of documentation which are useful resources which goes well with the lectures.

AUTHOR	TITLE	PUBLISHER
G. B. Thomas, Jr. et.al	Thomas Calculus (12th Edition)	Pearson (2010)
A. Gut	Probability: A Graduate Course	Springer (2005)
S.L. Sobolev	Partial Differential Equations of Mathematical Physics	Addison Wesley (2019)
W. A. Strauss	Partial Differential Equations - An Introduction	Wiley (2008)
R. E. Walpole, et. al	Probability and Statistics for Engineers & Scientists	Pearson (2012)
M. L. Boas	Mathematical Methods in the Physical Sciences (3rd Edition)	Wiley (2006)
K. F. Riley, et. al	Mathematical Methods for Physics and Engineering (3rd Edition)	Cambridge (2006)
G. F. Simmons	Differential Equations with Applications and Historical Notes (3rd Edition)	CRC Press (2017)
E. Kreyszig	Advanced Engineering Calculus (9th Edition)	Wiley (2011)
D. C. Montgomery	Applied Statistics and Probability for Engineers (3rd Edition)	Wiley (2003)
J. F. James	A Students Guide to Fourier Transform	Cambridge (2011)
J. Crank	Mathematics of Diffusion	Oxford (1975)
A. Sommerfeld	Partial Differential Equations in Physics	Academic Press (1949)
E. Cinlar	Probability and Stochastics	Springer (2010)
H. C. Berg	Random Walks in Biology	Princeton (1983)

Content and Unit Distribution

• The content and unit distribution of the lecture is as follows where a unit is defined as 45 min lecture.

TOPIC	UNITS	SELF STUDY
First-Order Ordinary Differential Equations	2	4
Second-Order Ordinary Differential Equations	4	8
Higher-Order Ordinary Differential Equations	2	4
Systems of ODEs	4	8
Special Functions for ODEs	2	4
Laplace Transform	4	8
Linear Algebra I - Fundamentals	2	4
Eigenvalue Problems	4	8
Vector Differential Calculus	4	8
Vector Integral Calculus	2	4
SUM	30	60

Lecture Table of Contents

The structure of the M.Sc Higher Mathematics I can be grouped into three (3) parts:

1. Ordinary Differential Equations (ODEs)
2. Linear Algebra
3. Vector Calculus

Below is the detailed structure of the lecture.

• First-Order Ordinary Differential Equations
  • Introduction to Modelling
    • Initial Value Problem
  • Separable ODEs
    • Reduction to Separable Form
  • Exact ODEs
    • Integrating Factors
  • Linear ODEs
    • Introduction
      • Homogeneous Linear ODE
      • Non-Homogeneous Linear ODE
• Second-Order Ordinary Differential Equations
  • Introduction
    • Superposition Principle
    • Initial Value Problem
    • Reduction of Order
  • Homogeneous Linear ODEs
    • A Study of Damped System
      • Case III: Under-Damping
    • Euler-Cauchy Equations
    • Non-homogeneous ODEs
      • Method of Undetermined Coefficients
      • Step 1: General Solution of the Homogeneous ODE
      • Step 2: Solution of the non-Homogeneous ODE
      • Step 3. Solution of the initial value problem.
      • Step 1.General solution of the homogeneous ODE
      • Step 2.Solution $y_{p
      • Step 3. Solution of the initial value problem
      • Step 3. Solution of the initial value problem
    • A Study of Forced Oscillations and Resonance
      • Solving the Non-homogeneous ODE
    • Solving Electric Circuits
      • Solving the ODE for the Current
      • Case I
      • Case II
      • Case III
      • Step 1. General solution of the homogeneous ODE
      • Step 2. Particular solution $I[p]$
    • Forced Oscillations
• Higher-Order Ordinary Differential Equations
  • Homogeneous Linear ODEs
    • Superposition and General Solution
    • General solution
    • Particular solution
    • Wronskian: Linear Independence of Solutions
    • Homogeneous Linear ODEs with Constant Coefficients
      • Distinct Real Roots
      • Simple Complex Roots
      • Multiple Real Roots
      • Multiple Complex Roots
    • Non-Homogeneous Linear ODEs
      • Step 1
      • Step 2
      • Step 3
    • Application: Modelling an Elastic Beam
      • Problem Description
      • Boundary Conditions
      • Solution Derivation
• Systems of ODEs
  • Introduction
    • System of ODEs as Models in Engineering
      • Setting Up the Model
      • General Solution
      • Use of initial conditions
      • Answer
      • Setting up the mathematical model
      • General Solution
    • Conversion of an n-th Order ODE to a System
    • Linear Systems
  • Constant-Coefficient Systems
    • Phase Plane Method
    • Critical Points of the System
      • Five Types of Critical Points
  • Criteria for Critical Points & Stability
  • Qualitative Methods for Non-Linear Systems
    • Linearisation of Non-Linear Systems
      • Setting Up the Mathematical Model
      • Critical Points ($pm2gpin,,0$) and Linearisation
      • Critical Points ($pm(2n -1)gpi,,0$) and Linearisation
• Special Functions for odes
  • Introduction
  • Power Series Method
  • Legendre’s Equation
    • Legendre Polynomials ($fnr{P[n]
      • Polynomial Solutions
  • Extended Power Series: Frobenius Method
    • Indicial Equation
    • Typical Applications
  • Bessel’s Function
    • Bessel Functions ($J[n]$) for Integers
    • Bessel Functions of the super{2
• Laplace Transform
  • Introduction
  • First Shifting Theorem (s-Shifting)
    • Replacing s by s - a in the Transform
  • Transforming Derivatives and Integrals
    • Laplace Transform a Function Integral
    • Differential Equations with Initial Values
  • Unit Step Function (t - Shifting)
    • Unit Step Function (Heaviside Function)
    • Time Shifting (t-Shifting): Replacing t by t - a in f(t)
  • Dirac Delta Function
  • Convolution
• Linear Algebra I - Fundamentals
  • Introduction
  • Matrices and Vectors
    • Addition and Scalar Multiplication
    • General Concepts and Notations
    • Vectors
  • Matrix Multiplication
  • Solutions to Linear Systems
    • Principles of Existence and Uniqueness
  • Second and Third Order Determinants
  • Linear Independence
    • Linear Independence and Dependence of Vectors
    • Vector Space
  • Solution of Linear Systems
  • Inverse of a Matrix
    • Gauss-Jordan Method
    • Trace of a Matrix
• Eigenvalue Problems
  • Introduction
  • The Eigenvalue Problem
    • Determining Eigenvalues and Eigenvectors
    • The Process of Finding Eigenvalues and Eigenvectors
  • Eigenvalue Applications
  • Symmetric, Skew-Symmetric and Orthogonal Matrices
    • Necessary Definitions
    • Orthogonal Transformations and Matrices
  • Eigenbases, Diagonalisation and Quadratic Forms
    • Similarity of Matrices and Diagonalisation
    • Quadratic Forms and Transformation to Principle Axis
  • Complex Matrices
    • Eigenvalues of Complex Matrices
• Vector Differential Calculus
  • Vectors in 2D Space
    • Vector Components
    • Addition and Scalar Multiplication
  • Inner Product
  • Vector Product
    • Scalar Triple Product
  • Vector Calculus: Derivatives
    • Vector Calculus
  • Theory of Curves
    • Tangent to a Curve
    • Length of a Curve
    • Arc Length s of a Curve
    • The Gradient of a Scalar Field
    • Directional Derivative
    • Gradient as A Vector Normal
    • The Gradient of a Scalar Field
  • Divergence of a Vector Field
  • Curl of a Vector Field
• Vector Integral Calculus
  • Introduction
  • Line Integrals
    • Defining and Evaluating Line Integrals
  • Path Independence of Line Integrals

(-DTMc 2025)