Higher Mathematics 1

Prof. Dr.-Ing. Dieter Kraus
   
1. Basics (pdf)
   1.1 Set Theory, Set Operations and Mappings
        1.1.1 Special Sets
        1.1.2 Set Operations
        1.1.3 Mappings
   1.2 Real Numbers
        1.2.1 Natural Numbers
        1.2.2 Integers
        1.2.3 Rational Numbers
        1.2.4 Mathematical Methods of Proof
        1.2.5 Irrational Numbers
        1.2.6 Axioms of der Real Numbers
        1.2.7 Applications
   1.3 Number Systems
   1.4 Complex Numbers

2. Lineare Algebra (pdf)
   2.1 Vektors in R3
        2.1.1 Vector Addition
        2.1.2 Vector Multiplication with a Scalar
        2.1.3 Modulus of a Vector
        2.1.4 Vectors in a Coordinate System
        2.1.5 Angle between Vectors
        2.1.6 Dot Product
        2.1.7 Vector Product
        2.1.8 Parallelepipedial Product
   2.2 Linear Spaces
   2.3 Matrices
        2.3.1 Addition, Subtraction, Multiplication with a Scalar
        2.3.2 Matrix Multiplication
        2.3.3 Transposed Matrix
        2.3.4 Inverse Matrix
        2.3.4 Symmetric, skew symmetric, orthogonal Matrices
   2.4 Linear Mappings
        2.4.1 Construction of the Matrix Representation of Linear Mappings
        2.4.2 Gram-Schmidt Orthogonalization
        2.4.3 Coordinate Transformation
   2.5 Linear Equation Systems
        2.5.1 Gaussian Elimination
        2.5.2 Geometrical Analysis
        2.5.3 Numerical Errorsr
        2.5.4 Ill-conditioned Matrix
        2.5.5 Calculation of the inverse Matrix
        2.5.6 Solvability of Linear Equation Systems
   2.6 Determinants
   2.7 Eigenvalues, Eigenvectors
   2.8 Quadratic Forms, Quadratic Polynomials

3. Functions, Limits, Continuity (pdf)
   3.1 Fundamental Terms
   3.2 Elementary Funktions
        3.2.1 Polynomial Functions
        3.2.2 Rational Functions
        3.2.3 Exponential- und Logarithm Functions
        3.2.4 Trigonometric Functions
        3.2.5 Hyperbolic Functions
   3.3 Sequences and Limits
        3.3.1 Introduction
        3.3.2 Convergence of Sequences
        3.3.3 Calculation Rules for convergent Sequences
        3.3.4 Nested Intervals (bisection method)
        3.3.5 Infinite Series
   3.4 Limits with Functions, Continuity
        3.4.1 Limits
        3.4.2 Continuity

4. Differential Calculus (pdf)
   4.1 Basics
        4.1.1 Derivative of differentiable Function
        4.1.2 Derivative of some elementary Functions
        4.1.3 Higher Derivatives
   4.2 Applications of Differential Calculus
        4.2.1 Properties of differentiable Functions
        4.2.2 Roots and Fixed Points
                4.2.2.1 Fixed Point Iteration
                4.2.2.2 Newton's Method
        4.2.3 Calculation of Limits, L'Hospitals's Rule
        4.2.4 Curve Sketching