Invariant points
An invariant point is a point that does not move. [...]
3D matrix transformations
You will need to be familiar with matrix transformations in [...]
The mean value of a function
Transformations of the mean value of a function [...]
The conic sections of a circle, parabola, ellipse and hyperbola
You will have met the circle and parabola in earlier [...]
Shortest distance of a point to a line
Finding the shortest distance of a point to a line [...]
Geometrical effects of conjugating a complex number
Example to try [...]
Modulus-argument form of a complex number
In this video tutorial you are introduced to the mod-arg [...]
Transposed and symmetric matrices
In this video I introduce you to the methods of [...]
Relationships between the roots and coefficients of a cubic equation
Finding a cubic equation based on roots of [...]
Relationships between the roots and coefficients of a quadratic equation
In these videos I show you how the sum and [...]
Volume of revolution about the x-axis generated between curves
I now extend the the basic volume of revolution about [...]
Volume of Revolution about the y-axis generated between curves
I now extend the the basic volume of revolution about [...]
Volume of Revolution about the y-axis
I now introduce you to the volume of revolution about [...]
Exam Questions – Matrix proofs
Please note: The matrix proof by induction is on [...]
Solving quartic equations
A quartic equation can have 4 real roots or 2 [...]
Solving cubic equations
A cubic equation can have 3 real roots or 1 [...]
Solving quadratic equations with complex roots
If a quadratic equation has complex roots then they are [...]
Square roots of a complex number
In this video I show you how to find the [...]
Cartesian form of a line
Using the vector parametric form for a line, I show [...]
Expressing inverse hyperbolic functions as natural logarithms
Inverse hyperbolic functions can be expressed in terms of natural [...]
Hyperbolic identities
In this video I show you 3 basic hyperbolic identities [...]
Solving equations using inverse and exponential functions
In this video I show you how to work solve [...]
Graphs of sinh(x), cosh(x) and tanh(x)
In this video I show you how you can use [...]
Definitions
Hyperbolic functions are similar to trigonometric functions. In this video [...]
Sketching curves the curve r = asin 2θ
I now introduce you to plotting the curve r=a sin2θ. [...]
Sketching curves the cardioid r = a (1+cosθ)
Now I show you a very common curve which you [...]
Sketching curves a circle and arc
Polar equations of circles and arcs. [...]
Sketching polar curves a half-line
I now move on to another common curve, the half-line. [...]
Sketching polar curves a spiral
I now introduce you to plotting some common types of [...]
Converting the equation of a Cartesian curve to polar form
I now work the other way round and show you [...]
Converting the equation of a polar curve to Cartesian form
In this video I now show you how to convert [...]
Converting polar coordinates to Cartesian coordinates
I now run over three examples where you have to [...]
Converting Cartesian coordinates to polar coordinates
I would encourage you to look at or even try [...]
Defining the position of a point
You would have used the cartesian system to define the [...]
Further series
In previous videos I have shown you the series expansions [...]
Series expansion for ln(1+x)
In this video I show you how to derive the [...]
Series expansion for sin(x) and cos(x)
In this video I show you how to derive the [...]
Series expansion for ex
In this video I show you how to derive the [...]
Maclaurin’s series expansion
Maclaurin's series can be used to find expansions of functions [...]
Using complex numbers to represent regions on an Argand diagram
In this tutorial I look at the locus of a [...]
The locus of a point moving along a half-line
In this tutorial I look at the locus of a [...]
The locus of a point moving in a circle
In this series of videos I look at the locus [...]
Multiplication and division rules for mod and argument of two complex numbers
Multiplication rule In this video I prove to you the [...]
Method of differences
In this video I show you a method known as [...]
Fractional inequalities
In this video I show you how to solve fractional [...]
Matrix proofs
Example 1 In this video I show you how to [...]
Divisibility and multiple test proofs
Proof that an expression is divisible by a certain integer [...]
Proof for other series
Example 1 In the following videos I show you how [...]
Proof of the sum of the series ∑r³
In this video I show you how to use mathematical [...]
Proof of the sum of the series ∑r²
In this video I show you how to use mathematical [...]
Proof of the sum of the series ∑r
In this video I show you how to use mathematical [...]
Using known formulae to sum more complex series
In this video I now combine the various summations for [...]
Sum of the cubes of the first n natural numbers ∑r3
In this video I now introduce you to the standard [...]
Sum of the squares of the first n natural numbers ∑r2
In this video I now introduce you to the [...]
Combinations of transformations
I now show you how we can combine two or [...]
How well do you know your transformations?
In this video I test you on compiling transformation matrices. [...]
Linear transformations – enlargement
In this series of tutorials I show you how we [...]
Linear transformations – reflections
In this series of tutorials I show you how we [...]
Linear transformations – rotations
In this series of tutorials I show you how we [...]
Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)
In this series of videos I continue to extend [...]
Identity and Inverse of a 2×2 matrix
There are two matrices which are very important and are [...]
Matrix multiplication
In this tutorial I show you how to multiply two [...]
Addition and subtraction and multiplying a matrix by a scalar
In this tutorial I show you how to do addition [...]
Introduction and dimension of a matrix
In this tutorial, I show you what a matrix is [...]
Solving problems with complex numbers
In this tutorial I show you how to solve problems [...]
Modulus and argument of a complex number
In this tutorial you are introduced to the modulus and [...]
Argand diagrams
The Argand diagram is used to represent complex numbers. In [...]
Division of a complex number by a complex number
I now show you how we can use complex conjugates [...]
Complex conjugates
What do we mean by a complex conjugate and complex [...]
Addition, subtraction and multiplying complex numbers and simplifying powers of i
These are the basic skills of handling complex numbers that [...]
Real and imaginary numbers
In this video you are introduced to complex numbers. I [...]
Closest point to a line and shortest distance from the origin
In this video I show you how to find the [...]
Intersecting and skew lines
If two lines are not parallel then they will either [...]
Parallel lines
In this video you are shown the conditions for two [...]
Angle between two lines
In this video you are shown how to find the [...]
Vector equation of a line
This tutorial will show you how to get to any [...]
Perpendicular vectors
The scalar product can be used to test if two [...]
Scalar product
In this video I show you how the scalar product [...]
Volume of revolution about the x-axis
In the first video in this series, I show you [...]