Higher Mathematics: Comprehensive University-
Level Notes
Chapter 1: Calculus
1.1 Limits and Continuity
• Definition of a limit
• One-sided limits
• Properties of limits
• L’Hôpital’s rule
• Continuity of functions
• Intermediate Value Theorem
1.2 Differentiation
• Definition of derivative
• Techniques: Product, Quotient, Chain rule
• Higher-order derivatives
• Implicit differentiation
• Applications: Tangents, Normals, Maxima and Minima
1.3 Integration
• Indefinite and definite integrals
• Techniques: Substitution, Integration by parts, Partial fractions
• Improper integrals
• Applications: Area, Volume, Arc length, Surface area
1.4 Multivariable Calculus
• Partial derivatives
• Gradient, Divergence, Curl
• Multiple integrals (double, triple)
• Change of variables (Jacobian)
• Vector calculus theorems: Green’s, Stokes’, Divergence theorem
Chapter 2: Linear Algebra
2.1 Matrices
• Types: Square, Diagonal, Identity, Symmetric, Skew-symmetric
• Operations: Addition, Multiplication, Transpose, Inverse
• Determinants and properties
1
, 2.2 Systems of Linear Equations
• Gaussian elimination
• Cramer’s Rule
• Rank of a matrix
• Homogeneous and non-homogeneous systems
2.3 Vector Spaces
• Definition, subspaces
• Linear independence, basis, dimension
• Null space, column space
2.4 Eigenvalues and Eigenvectors
• Characteristic equation
• Diagonalization
• Applications: Stability analysis, Differential equations
Chapter 3: Differential Equations
3.1 Ordinary Differential Equations (ODE)
• First-order ODEs: Separable, Exact, Linear
• Higher-order linear ODEs
• Method of undetermined coefficients
• Variation of parameters
• Applications: Mechanical vibrations, population dynamics
3.2 Partial Differential Equations (PDE)
• Basic types: Heat, Wave, Laplace equations
• Method of separation of variables
• Boundary and initial conditions
Chapter 4: Complex Analysis
4.1 Complex Numbers
• Algebraic form, Polar form
• Euler’s formula
• De Moivre’s theorem
4.2 Analytic Functions
• Cauchy-Riemann equations
2
Level Notes
Chapter 1: Calculus
1.1 Limits and Continuity
• Definition of a limit
• One-sided limits
• Properties of limits
• L’Hôpital’s rule
• Continuity of functions
• Intermediate Value Theorem
1.2 Differentiation
• Definition of derivative
• Techniques: Product, Quotient, Chain rule
• Higher-order derivatives
• Implicit differentiation
• Applications: Tangents, Normals, Maxima and Minima
1.3 Integration
• Indefinite and definite integrals
• Techniques: Substitution, Integration by parts, Partial fractions
• Improper integrals
• Applications: Area, Volume, Arc length, Surface area
1.4 Multivariable Calculus
• Partial derivatives
• Gradient, Divergence, Curl
• Multiple integrals (double, triple)
• Change of variables (Jacobian)
• Vector calculus theorems: Green’s, Stokes’, Divergence theorem
Chapter 2: Linear Algebra
2.1 Matrices
• Types: Square, Diagonal, Identity, Symmetric, Skew-symmetric
• Operations: Addition, Multiplication, Transpose, Inverse
• Determinants and properties
1
, 2.2 Systems of Linear Equations
• Gaussian elimination
• Cramer’s Rule
• Rank of a matrix
• Homogeneous and non-homogeneous systems
2.3 Vector Spaces
• Definition, subspaces
• Linear independence, basis, dimension
• Null space, column space
2.4 Eigenvalues and Eigenvectors
• Characteristic equation
• Diagonalization
• Applications: Stability analysis, Differential equations
Chapter 3: Differential Equations
3.1 Ordinary Differential Equations (ODE)
• First-order ODEs: Separable, Exact, Linear
• Higher-order linear ODEs
• Method of undetermined coefficients
• Variation of parameters
• Applications: Mechanical vibrations, population dynamics
3.2 Partial Differential Equations (PDE)
• Basic types: Heat, Wave, Laplace equations
• Method of separation of variables
• Boundary and initial conditions
Chapter 4: Complex Analysis
4.1 Complex Numbers
• Algebraic form, Polar form
• Euler’s formula
• De Moivre’s theorem
4.2 Analytic Functions
• Cauchy-Riemann equations
2
