Ifis a group of order 6, the orders of each element may only be 1, 2, 3 or 6, since the order of an element must divide the order of the group. Ifhas an element of order 6, it is cyclic and is isomorphic to Supposehas no element of order 6, but has an elementof […]

## Testing Whether a Set Forms a Group

To test whether a set along with an operation on the members of the set forms a group, we must test the group properties in turn. These are G1: Closure – ifthenwhereis the group operation. G2: Associativity – ifthen G3: Identity – every group has an identity elementsatisfyingfor all G4: Inverses – ifthenwhereand Only one […]

## Differentiating Artan x n Times

then we can findby writingand finding Findingbecomes the problem of finding Write In general hence

## Transforming Differential Equations

Differential equations may be transformed by a change of variables, making them simpler, and often easier to solve. Either the independent variable – usuallyoror the dependent variable, usuallymay be transformed, or both. The transformation must be chosen carefully, since not all transformations will make the equation simpler. For example, suppose we have to solve the […]

## The Integrating Factor Method of Solving First Order Differential Equations

The differential equationsis separable We can write this equation asIntegrating both sides gives where Some first order differential equations are not separable. Often the most suitable way to solve it is the integrating factor method, which can be used to solve equations of the form If we multiply both sides by the integrating factor,the equation […]

## Solving Second Order Constant Coefficient Linear Differential Equations

A second order constant coefficient linear differential equation is any equation of the form whereandare constants. Solving these involves Finding a solutionof the homogeneous equationWe may assume a solution of the formobtaining the ‘characteristic’ equationIf this equation has two distinct real rootsandthenIf the equation has a single repeated rootthenIf the equation has complex rootsthen Now […]

## Induction

Induction is a method of proof which relies on a statement P(m) being true for a certain member of a sequence and seeks to prove the statementtrue for succeeding values with the aid of some relationship between consecutive terms of the sequence. With the initial statement P(m), we take the induction step: Assumingis true for […]

## Finding the Volume of a Pyramid Using Vectors

The volume of a pyramid is In three dimensional space all the vertices of the pyramid can be given coordinates. The bass area can be expressed as the cross product of the vectorsand The To find the height of the pyramid, notice that the cross product of two vectors is perpendicular to both vectors.anddefine the […]

## Finding a Taylor Series for the Solution to a Differential Equation Given Initial Conditions

If the exact solution to a differential equation cannot be found one method of solving the equation is by repeated differentiation to find a Taylor series for the solution. In general the series will involve the introduction of constants, but these can be found if we have some initial or boundary conditions for the solution. […]

## de Moivre’s Theorem

De Moivre‘s theorem states that forwhere The theorem is easy to prove using the relationship Raising both sides of this expression to the power ofgives The theorem is useful when deriving relationships between trigonometric functions. For example, we can obtain polynomial expressions for sin n %theta and cos n %theta for any n using de […]