Math

This is Math Lecture Notes

MH1810

I Algebra 1

1 Complex Numbers

  • 1.1 The set of real numbers \(\mathbb{R}\) & its subsets 3
  • 1.2 Complex Numbers 3
  • 1.3 Argand diagram & Polar representation 5
  • 1.4 Operations on complex numbers 8
  • 1.5 The Fundamental Theorem of Algebra 13
  • 1.6 De Moivre’s Theorem 15
  • 1.7 Finding nth roots of \(z = r(\cos \alpha +i\sin \alpha)\) 18

2 Vectors

  • 2.1 Geometrical Representation of Vectors 25
  • 2.2 Vector Addition & Scalar Multiplication 25
  • 2.3 Vectors in Coordinate System 27
  • 2.4 Lines and Planes 38

3 Matrices

  • 3.1 Matrix Notation and Terminology 45
  • 3.2 Arithmetic Operations of Matrices 49
  • 3.3 Transpose 54
  • 3.4 Matrix Inverse 54
  • 3.5 Power 59
  • 3.6 Determinants 60
  • 3.7 Adjoint of A (Optional) 67
  • 3.8 Cramer’s Rule 68
  • 3.9 Proofs (Optional) 71

II Calculus 73

4a Limits and Continuity

  • 4.1 Limit of a Function at a Point 75
  • 4.2 One-sided Limit of a Function at a Point 77
  • 4.3 Infinite Limit 80
  • 4.4 Limits at Infinity 81
  • 4.5 Limit Theorems 82
  • 4.6 Continuity 84
  • 4.7 One-sided Continuity 87
  • 4.8 Properties on Continuity 87

4b Limits and Continuity

  • 4.9 Techniques in Finding Limits 88
  • 4.10 One-sided Limits 91
  • 4.11 Limit Laws for Infinite Limits 93
  • 4.12 Evaluation of Limits at Infinity 94
  • 4.13 Continuous Functions 98
  • 4.14 The Intermediate Value Theorem 101
  • 4.15 The Extreme Value Theorem 103

5a Differentiation

  • 5.1 Derivatives 105
  • 5.2 Differentiation Rules 113
  • 5.3 Implicit Differentiation and Differentiation of Inverse 121

5b Differentiation

  • 5.4 Problems on Rate of Change 125
  • 5.5 Linearization 127
  • 5.6 Newton’s Method 130
  • 5.7 Closed Interval Method 134

5c Differentiation

  • 5.8 Mean Value Theorem 135
  • 5.9 Maximum and Minimum Problems 140
  • 5.10 Second Derivative and the Nature of Extrema 142

6a Integration

  • 6.1 Antiderivatives & Indefinite Integral 147
  • 6.2 The Definite Integral and Area Under a Curve. 150
  • 6.3 The Fundamental Theorem of Calculus 156

6a Integration

  • 6.4 Techniques of Integration 164
  • 6.5 Integration of Rational Functions 168
  • 6.6 Improper Integrals 176

6c Integration

  • 6.7 Area and Volume 182
  • 6.8 Numerical Integration 188