The cross product of two vectorsandis also a vector, at right angles to the other two with magnitudewhereis the angle between the two vectors. The cross product of the two vectors can be written as the determinant of a matrix: If andthen For example if and then As shown above, if the order of the […]

## Basic Maclaurin Series

Maclaurin series are closely related to Taylor series, being the Taylor series expansion of a function f(x) about x=0. The Maclaurin series of some functions are show below. Higher order terms which include factors called Bernoulli numbers. Higher order terms again include factors called Bernoulli numbers. forHigher order terms include factors called Euler numbers. forHigher […]

## Improved Euler Formulae for Solving First Order Differential Equations Numerically

Several methods exists for finding better numerical solutions to first order differential equations of the formthan Euler’s simple formula One method uses forwards and backwards Euler formulae to derive a ‘central’ formula. Subtracting these gives Theterm means that the error at each iteration is of the order ofThe error for the simple Euler forwards or […]

## Finding Cartesian Equations for Curves Given in Polar Coordinates

Typically a curve is given in polar coordinateswithas a function ofIt is often quite simple to write this in cartesian coordinatesby making the substitutionsand simplifying the resulting expression. Example: On substituting these, the equation becomes Subtract the terms on the right hand side to give We can complete the square for both the‘s and‘s to […]

## The Euler Formula for Numerical Solutions of First Order Differential Equations

The Euler (forward) formula is the simplest numerical method for estimating values of a function at a point. It can be applied to any problem of the formsubject to the conditionThe Euler formula is iterative. The iteration formulae are Similar backwards iteration formulae exist. The Euler formula is one of the least accurate numerical methods. […]

## Basic Taylor Series

The Taylor series of a function about a pointis given by Taylor series have many purposes, from approximating a function, allowing approximate solutions of equations to be found, in complex analysis, analysis, and many more areas. Taylor series can be multiplied or divided to find the Taylor series of products or quotients of functions, inverted […]

## Finding Inverses of 3×3 Matrices

Finding the inverse of a 3×3 matrix is quite involved. The steps are: 1. Form the adjoint matrix. 2. Permute the signs according to 3. Take the transpose. 4. Find the determinantof the original matrix and divide the matrix by it, which means multiplying the matrix by a factor Example: Find the inverse of We […]

## The Image of a Line Under Transformation By a Matrix

A matrix is a linear transformation, which means that it sends a line to a line. To find the image of the lineon transformation by the matrixwe represent the line by the vectorand multiply the vector by the matrix: From this we see that (1) (2) Rearrange (1) to makethe subject and substitute into (2): […]

## Properties of Groups – A Summary

A groupis a nonempty set of elements . Ifthen along with a group multiplication operation · (called the product) satisfying the following four conditions 1. Closure. Ifthenis also in 2. Associativity. The group multiplication is associative, a · (b · c) = (a · b) · c 3. The identity, denotedis a member of the […]

## Roots and Powers of Complex Numbers

It is a known fact thatand in general a square root always has two roots, one positive and one negative. Alsohas four possible answers, since Briefly, if with take the nth root of a number we obtain n answers: This is illustrated on the Argand diagram below which shows ten tenth roots of 1. To […]