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This document contains a 146 page PDF revision guide on “Mathematics for Engineers and Scientists,” which is specific to the Engineering course at Durham University. It has been written by a Durham University student who has completed the course and achieved a 1st. The notes have been tailored for students who are looking to revise for exams or complete coursework. The document covers the following topics:

**Complex Numbers**- Number Classification
- Fundamental Theorem of Algebra
- Definitions
- Addition and Subtraction
- Complex Conjugation
- Multiplication
- Modulus
- Division
- Polar Coordinates
- Euler Formula
- De Moivre’s’ Theorem
- Roots and Powers of Complex Numbers
- The General Expression
- Polynomial Expressions
- Logarithms
- Some Geometry
- Application – LCR Circuits
- Application – Mapping and 2D flow

**Calculus, Limits and Series**- Limits Introduction
- Calculus of Limits
- Limits of a function
- The pinching/squeezing theorem
- Continuity
- Differentiation and differentiability
- L’Hopital’s Rule
- Taylor Series
- Leibnetz’ rule
- Calculus of Taylor’s Series
- Integrating and Differentiating
- Newton-Raphson Method

**Vectors**- Notation and definitions of vectors
- Vector addition
- Scalar multiplication
- Laws of addition, subtraction and scalar multiplication
- Bases and components
- Scalar (dot) product
- Equation of a straight line
- Equation of a plane
- Parametric equation of a plane
- Work done
- Intersecting planes
- Vector (cross) product
- Scalar triple product
- Vector equation of a straight line
- Distance between 2 lines
- Differentiation of vectors
- Vector mechanisms

**Functions of Several Variables**- Partial differentiation
- Chain rule
- Higher dimensions
- Directional derivatives
- Tangent planes and normals to surfaces
- Div and Curl
- Critical Points
- Differential Equations
- General solutions and particular solutions
- 1st order ODE’s
- Homogeneous ODE’s
- Linear ODE’s
- Exact ODE’s
- Second order linear constant coefficient ODE’s

**Linear Algebra**- Matrices and Vectors Introduction
- Using matrices for linear algebra
- Gaussian elimination
- Determinants and inverses
- LU factorisation
- Application to electrical circuits

**Vector Spaces**- Vector spaces introduction
- Eigenvectors and eigenvalues
- Linear system of ODE’s
- Numerical Linear Algebra

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