JUST THE MATHS
UNIT NUMBER
8.5
VECTORS 5
(Vector equations of straight lines)
by
A.J.Hobson
8.5.1 Introduction
8.5.2 The straight line passing through a given point and
parallel to a given vector
8.5.3 The straight line passing through two given points
8.
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UNIT NUMBER
12.6
INTEGRATION 6
(Integration by partial fractions)
by
A.J.Hobson
12.6.1 Introduction and illustrations
12.6.2 Exercises
12.6.3 Answers to exercises
UNIT 12.6 - INTEGRATION 6
INTEGRATION BY PARTIAL FRACTIONS
12.6.1 INTRODUCTIO
JUST THE MATHS
UNIT NUMBER
12.4
INTEGRATION 4
(Integration by substitution in general)
by
A.J.Hobson
12.4.1
12.4.2
12.4.3
12.4.4
Examples using the standard formula
Integrals involving a function and its derivative
Exercises
Answers to exercises
UNIT 12.4
JUST THE MATHS
UNIT NUMBER
12.5
INTEGRATION 5
(Integration by parts)
by
A.J.Hobson
12.5.1 The standard formula
12.5.2 Exercises
12.5.3 Answers to exercises
UNIT 12.5 - INTEGRATION 5
INTEGRATION BY PARTS
12.5.1 THE STANDARD FORMULA
The technique to be disc
JUST THE MATHS
UNIT NUMBER
12.9
INTEGRATION 9
(Reduction formulae)
by
A.J.Hobson
12.9.1
12.9.2
12.9.3
12.9.4
Indenite integrals
Denite integrals
Exercises
Answers to exercises
UNIT 12.9 - INTEGRATION 9
REDUCTION FORMULAE
INTRODUCTION
For certain integrals
JUST THE MATHS
UNIT NUMBER
12.8
INTEGRATION 8
(The tangent substitutions)
by
A.J.Hobson
12.8.1
12.8.2
12.8.3
12.8.4
The substitution t = tan x
The substitution t = tan(x/2)
Exercises
Answers to exercises
UNIT 12.8 - INTEGRATION 8
THE TANGENT SUBSTITUTIONS
JUST THE MATHS
UNIT NUMBER
12.7
INTEGRATION 7
(Further trigonometric functions)
by
A.J.Hobson
12.7.1
12.7.2
12.7.3
12.7.4
Products of sines and cosines
Powers of sines and cosines
Exercises
Answers to exercises
UNIT 12.7 - INTEGRATION 7 - FURTHER TRIGONOM
JUST THE MATHS
UNIT NUMBER
14.2
PARTIAL DIFFERENTIATION 2
(Partial derivatives of order higher than one)
by
A.J.Hobson
14.2.1 Standard notations and their meanings
14.2.2 Exercises
14.2.3 Answers to exercises
UNIT 14.2 - PARTIAL DIFFERENTIATION 2
PARTIAL
JUST THE MATHS
UNIT NUMBER
12.10
INTEGRATION 10
(Further reduction formulae)
by
A.J.Hobson
12.10.1
12.10.2
12.10.3
12.10.4
12.10.5
12.10.6
Integer powers of a sine
Integer powers of a cosine
Walliss formulae
Combinations of sines and cosines
Exercises
Ans
JUST THE MATHS
UNIT NUMBER
14.3
PARTIAL DIFFERENTIATION 3
(Small increments and small errors)
by
A.J.Hobson
14.3.1
14.3.2
14.3.3
14.3.4
14.3.5
Functions of one independent variable - a recap
Functions of more than one independent variable
The logarithmic
JUST THE MATHS
UNIT NUMBER
14.6
PARTIAL DIFFERENTIATION 6
(Implicit functions)
by
A.J.Hobson
14.6.1
14.6.2
14.6.3
14.6.4
Functions of two variables
Functions of three variables
Exercises
Answers to exercises
UNIT 14.6 - PARTIAL DIFFERENTIATION 6
IMPLICIT
JUST THE MATHS
UNIT NUMBER
14.4
PARTIAL DIFFERENTIATION 4
(Exact dierentials)
by
A.J.Hobson
14.4.1
14.4.2
14.4.3
14.4.4
14.4.5
Total dierentials
Testing for exact dierentials
Integration of exact dierentials
Exercises
Answers to exercises
UNIT 14.4 - PART
JUST THE MATHS
UNIT NUMBER
14.5
PARTIAL DIFFERENTIATION 5
(Partial derivatives of composite functions)
by
A.J.Hobson
14.5.1
14.5.2
14.5.3
14.5.4
Single independent variables
Several independent variables
Exercises
Answers to exercises
UNIT 14.5 - PARTIAL
JUST THE MATHS
UNIT NUMBER
14.7
PARTIAL DIFFERENTIATION 7
(Change of independent variable)
by
A.J.Hobson
14.7.1 Illustrations of the method
14.7.2 Exercises
14.7.3 Answers to exercises
UNIT 14.7 - PARTIAL DIFFERENTIATON 7
CHANGE OF INDEPENDENT VARIABLE
14
JUST THE MATHS
UNIT NUMBER
14.8
PARTIAL DIFFERENTIATION 8
(Dependent and independent functions)
by
A.J.Hobson
14.8.1 The Jacobian
14.8.2 Exercises
14.8.3 Answers to exercises
UNIT 14.8 - PARTIAL DIFFERENTIATION 8
DEPENDENT AND INDEPENDENT FUNCTIONS
14.8.1
JUST THE MATHS
UNIT NUMBER
14.9
PARTIAL DIFFERENTIATION 9
(Taylors series)
for
(Functions of several variables)
by
A.J.Hobson
14.9.1 The theory and formula
14.9.2 Exercises
UNIT 14.9 - PARTIAL DIFFERENTIATION 9
TAYLORS SERIES FOR FUNCTIONS OF SEVERAL VARI
JUST THE MATHS
UNIT NUMBER
14.11
PARTIAL DIFFERENTIATION 11
(Constrained maxima and minima)
by
A.J.Hobson
14.11.1
14.11.2
14.11.3
14.11.4
The substitution method
The method of Lagrange multipliers
Exercises
Answers to exercises
UNIT 14.11 - PARTIAL DIFFER
JUST THE MATHS
UNIT NUMBER
12.1
INTEGRATION 1
(Elementary indenite integrals)
by
A.J.Hobson
12.1.1
12.1.2
12.1.3
12.1.4
The denition of an integral
Elementary techniques of integration
Exercises
Answers to exercises
UNIT 12.1 - INTEGRATION 1 - ELEMENTARY
JUST THE MATHS
UNIT NUMBER
12.3
INTEGRATION 3
(The method of completing the square)
by
A.J.Hobson
12.3.1 Introduction and examples
12.3.2 Exercises
12.3.3 Answers to exercises
UNIT 12.3 - INTEGRATION 3
THE METHOD OF COMPLETING THE SQUARE
12.3.1 INTRODUCTI
JUST THE MATHS
UNIT NUMBER
12.2
INTEGRATION 2
(Introduction to denite integrals)
by
A.J.Hobson
12.2.1 Denition and examples
12.2.2 Exercises
12.2.3 Answers to exercises
UNIT 12.2 - INTEGRATION 2
INTRODUCTION TO DEFINITE INTEGRALS
12.2.1 DEFINITION AND EXA
JUST THE MATHS
UNIT NUMBER
8.6
VECTORS 6
(Vector equations of planes)
by
A.J.Hobson
8.6.1 The plane passing through a given point and
perpendicular to a given vector
8.6.2 The plane passing through three given points
8.6.3 The point of intersection of a s
JUST THE MATHS
UNIT NUMBER
9.1
MATRICES 1
(Denitions & elementary matrix algebra)
by
A.J.Hobson
9.1.1
9.1.2
9.1.3
9.1.4
9.1.5
Introduction
Denitions
The algebra of matrices (part one)
Exercises
Answers to exercises
UNIT 9.1 - MATRICES 1
DEFINITIONS AND EL
JUST THE MATHS
UNIT NUMBER
9.2
MATRICES 2
(Further matrix algebra)
by
A.J.Hobson
9.2.1
9.2.2
9.2.3
9.2.4
9.2.5
9.2.6
Multiplication by a single number
The product of two matrices
The non-commutativity of matrix products
Multiplicative identity matrices
Ex
JUST THE MATHS
UNIT NUMBER
9.3
MATRICES 3
(Matrix inversion & simultaneous equations)
by
A.J.Hobson
9.3.1
9.3.2
9.3.3
9.3.4
9.3.5
9.3.6
Introduction
Matrix representation of simultaneous linear equations
The denition of a multiplicative inverse
The formul
JUST THE MATHS
UNIT NUMBER
9.4
MATRICES 4
(Row operations)
by
A.J.Hobson
9.4.1
9.4.2
9.4.3
9.4.4
Matrix inverses by row operations
Gaussian elimination (the elementary version)
Exercises
Answers to exercises
UNIT 9.4 - MATRICES 4 - ROW OPERATIONS
9.4.1 MA
JUST THE MATHS
UNIT NUMBER
9.5
MATRICES 5
(Consistency and rank)
by
A.J.Hobson
9.5.1
9.5.2
9.5.3
9.5.4
9.5.5
The consistency of simultaneous linear equations
The row-echelon form of a matrix
The rank of a matrix
Exercises
Answers to exercises
UNIT 9.5 - M
JUST THE MATHS
UNIT NUMBER
9.6
MATRICES 6
(Eigenvalues and eigenvectors)
by
A.J.Hobson
9.6.1
9.6.2
9.6.3
9.6.4
The statement of the problem
The solution of the problem
Exercises
Answers to exercises
UNIT 9.6 - MATRICES 6
EIGENVALUES AND EIGENVECTORS
9.6.1
JUST THE MATHS
UNIT NUMBER
9.7
MATRICES 7
(Linearly independent eigenvectors)
&
(Normalised eigenvectors)
by
A.J.Hobson
9.7.1
9.7.2
9.7.3
9.7.4
Linearly independent eigenvectors
Normalised eigenvectors
Exercises
Answers to exercises
UNIT 9.7 - MATRICES 7
JUST THE MATHS
UNIT NUMBER
9.9
MATRICES 9
(Modal & spectral matrices)
by
A.J.Hobson
9.9.1
9.9.2
9.9.3
9.9.4
Assumptions and denitions
Diagonalisation of a matrix
Exercises
Answers to exercises
UNIT 9.9 - MATRICES 9
MODAL AND SPECTRAL MATRICES
9.9.1 ASSUMP
JUST THE MATHS
UNIT NUMBER
9.8
MATRICES 8
(Characteristic properties)
&
(Similarity transformations)
by
A.J.Hobson
9.8.1
9.8.2
9.8.3
9.8.4
Properties of eigenvalues and eigenvectors
Similar matrices
Exercises
Answers to exercises
UNIT 9.8 - MATRICES 8
CHA