Mathematics for Engineers and Scientists Revision Guide

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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