# Mathematics for Chemistry/Print version

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# Table of contents

edit- Introduction
- Number theory
- Functions
- Units and dimensions
- Statistics
- Plotting graphs
- Complex numbers
- Trigonometry
- Vectors
- Matrices and determinants
- Differentiation
- Integration
- Some useful aspects of calculus
- Enzyme kinetics
- Some mathematical examples applied to chemistry
- Tests and exams
- Further reading

# Introduction

edit- This book was initially derived from a set of notes used in a university chemistry course. It is hoped it will evolve into something useful and develop a set of open access
**problems**as well as pedagogical material.

For many universities the days when admission to a Chemistry, Chemical Engineering, Materials Science or even Physics course could require the equivalent of A-levels in Chemistry, Physics and Mathematics are probably over for ever. The broadening out of school curricula has had several effects, including student entry with a more diverse educational background and has also resulted in the subject areas Chemistry, Physics and Mathematics becoming **disjoint** so that there is no co-requisite material between them. This means that, for instance, physics cannot have any advanced, or even any very significant mathematics in it. This is to allow the subject to be studied without any of the maths which might be first studied by the A-level maths group at the ages of 17 and 18. Thus physics at school has become considerably more descriptive and visual than it was 20 years ago. The same applies to a lesser extent to chemistry.

This means there must be an essentially *remedial* component of university chemistry to teach just the Mathematics and Physics which is needed and not too much, if any more, as it is time consuming and perhaps not what the student of Chemistry is most focused on. There is therefore also a need for a book Physics for Chemistry.

## Quantitative methods in chemistry

editThere are several reasons why numerical (quantitative) methods are useful in chemistry:

- Chemists need numerical information concerning reactions, such as how much of a substance is consumed, how long does this take, how likely is the reaction to take place.
- Chemists work with a variety of different units, with wildly different ranges, which one must be able to use and convert with ease.
- Measurements taken during experiments are not perfect, so evaluation and combination of errors is required.
- Predictions are often desired, and relationships represented as equations are manipulated and evaluated in order to obtain information.

# Numbers

edit- ==Numbers==

Real numbers come in several varieties and forms;

- Natural numbers are integers that are greater than or equal to zero.
- Integers are whole numbers used for counting indivisible objects, together with negative equivalents and zero,
*e.g.*42, -7, 0 - Rational numbers can always be expressed as fractions,
*e.g.*4.673 = 4673/1000. - Irrational numbers, unlike rational numbers, cannot be expressed as a fraction or as a definite decimal,
*e.g.*and

It is also worth noting that the imaginary unit and therefore complex numbers are used in chemistry, especially when dealing with equations concerning waves.

### Surds

editThe origin of surds goes back to the Greek philosophers. It is relatively simple to prove that the square root of 2 cannot be a ratio of two integers, no matter how large the integers may become. In a rather Pythonesque incident the inventor of this proof was put to death for heresy by the other philosophers because they could not believe such a pure number as the root of 2 could have this *impure* property.

(The original use of quadratic equations is very old, Babylon many centuries BC.) This was to allocate land to farmers in the same quantity as traditionally held after the great floods on the Tigris and Euphrates had reshaped the fields. The mathematical technology became used for the same purpose in the Nile delta.

When you do trigonometry later you will see that surds are in the trigonometric functions of the important symmetrical angles, *e.g.* and so they appear frequently in mathematical expressions regarding 3 dimensional space.

### Notation

editThe notation used for recording numbers in chemistry is the same as for other scientific disciplines, and appropriately called scientific notation, or *standard form*. It is a way of writing both very large and very small numbers in a shortened form compared to decimal notation. An example of a number written in scientific notation is

with being a coefficient termed the significand or the *mantissa*, and being an integer exponent. When written in decimal notation, the number becomes

.

Numbers written in scientific notation are usually normalised, such that only one digit precedes the decimal point. This is to make order of magnitude comparisons easier, by simply comparing the exponents of two numbers written in scientific notation, but also to minimise transcription errors, as the decimal point has an assumed position after the first digit. In computing and on calculators, it is common for the ("times ten to the power of") to be replaced with "E" (capital e). It is important not to confuse this "E" with the mathematical constant *e*.

Engineering notation is a special restriction of scientific notation where the exponent must be divisible by three. Therefore, engineering notation is not normalised, but can easily use SI prefixes for magnitude.

Remember that in SI, numbers do not have commas between the thousands, instead there are spaces, *e.g.* , (an integer) or . Commas are used as decimal points in many countries.

### Exponents

edit This section is a stub.You can help Wikibooks by expanding it. |

Consider a number , where is the base and is the exponent. This is generally read as “ to the ” or “ to the power of ”. If then it is common to say “ squared”, and if then “ cubed”. Comparing powers (exponentiation) to multiplication for positive integer values of n , it can be demonstrated that

, *i.e.* four lots of added together

, *i.e.* multiplied by itself four times.

For , the result is simply . For the result is .

can be reduced to multiplication like this if is an integer:

### Order of operations

editWhen an expression contains different operations, they must be evaluated in a certain order. Exponents are evaluated first. Then, multiplication and division are evaluated from left to right. Last, addition and subtraction are evaluated left to right. Parentheses or brackets take precedence over all operations. Anything within parentheses must be calculated first. A common acronym used to remember the order of operations is **PEMDAS**, for "Parentheses, Exponents, Multiplication, Division, Addition, Subtraction". Another way to remember this acronym is "Please Excuse My Dear Aunt Sally".

Keep in mind that negation is usually considered multiplication. So in the case of , the exponent would be evaluated first, then negated, resulting in a negative number.

Take note of this example:

If evaluated incorrectly (left-to-right, with no order of operations), the result would be 16. Three plus five gives eight, times two is 16. The correct answer should be 13. Five times two gives ten, plus three gives 13. This is because multiplication is solved before addition.

## Partial fractions

editPartial fractions are used in a few derivations in thermodynamics and they are good for practicing algebra and factorisation.

It is possible to express quotients in more than one way. Of practical use is that they can be collected into one term or generated as several terms by the method of partial fractions. Integration of a complex single term quotient is often difficult, whereas by splitting it up into a sum, a sum of standard integrals is obtained. This is the principal chemical application of partial fractions.

An example is

In the above must equal since the denominators are equal. So we set first to +1 giving . Therefore B = -1/2. If we set instead , therefore . So

We can reverse this process by use of a *common denominator*.

The numerator is , so it becomes

which is what we started from.

So we can generate a single term by multiplying by the denominators to create a common denominator and then add up the numerator to simplify. A typical application might be to convert a term to partial fractions, do some calculus on the terms, and then regather into one quotient for display purposes. In a factorised single quotient it will be easier to see where numerators go to zero, giving solutions to , and where denominators go to zero giving infinities.

A typical example of a meaningful infinity in chemistry might be an expression such as

The variable is the energy E, so this function is small everywhere, except near . Near a *resonance * occurs and the expression becomes infinite when the two energies are precisely the same. A molecule which can be electronically excited by light has several of these resonances.

Here is another example. If we had to integrate the following expression we would first convert to partial fractions:

so

let then

let then

therefore the expression becomes

Later you will learn that these expressions integrate to give simple expressions in terms of the natural logarithm.

### Problems

edit

### Answers

edit

## Polynomial division

editThis is related to partial fractions in that its principal use is to facilitate integration.

Divide out

like this

3x - 7 ----------------- x + 1 ) 3x^{2}-4x -6 3x^{2}+3x --------- 0 -7x -6 -7x -7 -------- 1

So our equation becomes

This can be easily differentiated, and integrated. If this is differentiated with the quotient formula it is considerably harder to reduce to the same form. The same procedure can be applied to partial fractions.

## Substitutions and expansions

editYou can see the value of changing the variable by simplifying

to

where

This is an example of simplification. It would actually be possible to differentiate this with respect to either or using only the techniques you have been shown. The algebraic manipulation involves *differentiation of a quotient* and the *chain rule*.

Evaluating gives

Expanding this out to the s and s would look ridiculous.

Substitutions like this are continually made for the purpose of having new, simpler expressions, to which the rules of calculus or identities are applied.

# Functions

edit**A reader has identified this chapter as an undeveloped draft or outline.**

You can help to develop the work, or you can ask for assistance in the project room.## Functions as tools in chemistry

edit### The quadratic formula

editIn order to find the solutions to the general form of a quadratic equation,

there is a formula

(Notice the line over the square root has the same

*priority*as a bracket. Of course we all know by now that is not equal to but errors of priority are among the most common algebra errors in practice).There is a formula for a cubic equation but it is rather complicated and unlikely to be required for undergraduate-level study of chemistry. Cubic and higher equations occur often in chemistry, but if they do not factorise they are usually solved by computer.

Solve:

Notice the

*scope*or*range*of the bracket.Notice here that the variable is a

*concentration,*not the ubiquitous .# Units and dimensions

edit- ==Units, multipliers and prefixes==

It is usually necessary in chemistry to be familiar with at least three systems of units, Le Système International d'Unités (SI), atomic units as used in theoretical calculations and the unit system used by the experimentalists. Thus if we are dealing with the ionization energy, the units involved will be the Joule (J), the Hartree (E

_{h}, the atomic unit of energy), and the electron volt (eV).These units all have their own advantages;

- The SI unit should be understood by all scientists regardless of their field.
- The atomic unit system is the natural unit for theory as most of the fundamental constants are unity and equations can be cast in dimensionless forms.
- The electron volt comes from the operation of the ionization apparatus where individual electrons are accelerated between plates which have a potential difference in Volts.

An advantage of the SI system is that the dimensionality of each term is made clear as the fundamental constants have structure. This is a complicated way of saying that if you know the dimensionality of all the things you are working with you know an awful lot about the mathematics and properties such as scaling with size of your system. Also, the same system of units can describe both the output of a large power station (gigaJoules), or the interaction of two inert gas atoms, (a few kJ per mole or a very small number of Joules per molecule when it has been divided by Avogadro's number).

In SI the symbols for units are lower case unless derived from a person's name,

*e.g.*ampere is A and kelvin is K.SI base units Name Symbol Quantity metre **m**length kilogram **kg**mass second **s**time ampere **A**electric current kelvin **K**thermodynamic temperature candela **cd**luminous intensity mole **mol**amount of substance Derived units used in chemistry Quantity Unit Name Symbol Area m ^{2}Volume m ^{3}Velocity m s ^{-1}Acceleration m s ^{-2}Density kg m ^{-3}Entropy J mol ^{-1}K^{-1}Force kg m s ^{-2}newton N Energy N m joule J Pressure N m ^{-2}pascal Pa Frequency s ^{-1}hertz Hz Approved prefixes for SI units Prefix Factor Symbol atto 10 ^{-18}a femto 10 ^{-15}f pico 10 ^{-12}p nano 10 ^{-9}n micro 10 ^{-6}μ milli 10 ^{-3}m centi 10 ^{-2}c deci 10 ^{-1}d kilo 10 ^{3}k mega 10 ^{6}M giga 10 ^{9}G tera 10 ^{12}T peta 10 ^{15}P exa 10 ^{18}E Note the use of capitals and lower case and the increment on the exponent being factors of 3. Notice also centi and deci are supposed to disappear with time leaving only the powers of 1000.

## Conversion factors

editThe , (sometimes call caret or hat), sign is another notation for

*to the power of*. E means*times 10 to the power of*, and is used a great deal in computer program output.### Energy

editAn approximation of how much of a chemical bond each energy corresponds to is placed next to each one. This indicates that light of energy 4 eV can break chemical bonds and possibly be dangerous to life, whereas infrared radiation of a few cm

^{-1}is harmless.- 1 eV = 96.48530891 kJ mol
^{-1}(Near infrared), approximately 0.26 chemical bonds - 1 kcal mol
^{-1}= 4.184000000 kJ mol^{-1}(Near infrared), approximately 0.01 chemical bonds - 1 MHz = 0.399031E-06 kJ mol
^{-1}(Radio waves), approximately 0.00 chemical bonds - 1 cm
^{-1}= 0.01196265819 kJ mol^{-1}(Far infrared), approximately 0.00 chemical bonds

Wavelength, generally measure in nanometres and used in UV spectroscopy is defined as an inverse and so has a reciprocal relationship.

### Length

editThere is the metre, the Angstrom (10

^{-10}m), the micron (10^{-6}m), the astronomical unit (AU)and many old units such as feet, inches and light years.### Angles

editThe radian to degree conversion is 57.2957795130824, (i.e. a little bit less than 60, remember your equilateral triangle and radian sector).

### Dipole moment

edit1 Debye = 3.335640035 Cm x 10

^{-30}(coulomb metre)### Magnetic Susceptibility

edit1 cm

^{3}mol^{-1}= 16.60540984 JT^{-2}x 10^{30}(joule tesla^{2})### Old units

editOccasionally, knowledge of older units may be required. Imperial units, or convert energies from BTUs in a thermodynamics project, etc.

In university laboratory classes you would probably be given material on the

*Quantity Calculus*notation and methodology which is to be highly recommended for scientific work.A good reference for units, quantity calculus and SI is: I. Mills, T. Cuitas, K. Homann, N. Kallay, K. Kuchitsu,

*Quantities, units and symbols in physical chemistry, 2nd edition*, (Oxford:Blackwell Scientific Publications,1993).## Greek alphabet

edit**This section is a stub.**

You can help Wikibooks by expanding it.## Unit labels

editThe labelling of tables and axes of graphs should be done so that the numbers are dimensionless,

*e.g.*temperature is ,and energy mol / kJ

*etc*.This can look a little strange at first. Examine good text books like Atkins Physical Chemistry which follow SI carefully to see this in action.

The hardest thing with conversion factors is to get them the right way round. A common error is to divide when you should be multiplying, also another common error is to fail to raise a conversion factor to a power.

1 eV = 96.48530891 kJ mol

^{-1}1 cm

^{-1}= 0.01196265819 kJ mol^{-1}To convert eV to cm

^{-1}, first convert to kJ per mole by multiplying by 96.48530891 / 1. Then convert to cm^{-1}by multiplying by 1 / 0.01196265819 giving 8065.540901. If we tried to go directly to the conversion factor in 1 step it is easy to get it upside down. However, common sense tell us that there are a lot of cm^{-1}s in an eV so it should be obviously wrong.1 inch = 2.54 centimetres. If there is a surface of nickel electrode of 2 * 1.5 square inches it must be 2 * 1.5 * 2.54

^{2}square centimetres.To convert to square metres, the SI unit we must divide this by 100 * 100 not just 100.

### Dimensional analysis

edit**This section is a stub.**

You can help Wikibooks by expanding it.The technique of adding unit labels to numbers is especially useful, in that analysis of the units in an equation can be used to double-check the answer.

## An aside on scaling

editOne of the reasons powers of variables are so important is because they relate to the way quantities scale. Physicists in particular are interested in the way variables scale in the limit of very large values. Take cooking the turkey for Christmas dinner. The amount of turkey you can afford is linear, (power 1), in your income. The size of an individual serving is quadratic, (power 2), in the radius of the plates being used. The cooking time will be something like cubic in the diameter of the turkey as it can be presumed to be linear in the mass.

(In the limit of a very large turkey, say one the diameter of the earth being heated up by a nearby star, the internal conductivity of the turkey would limit the cooking time and the time taken would be exponential. No power can go faster / steeper than exponential in the limit. The series expansion of goes on forever even though gets very small.)

Another example of this is why dinosaurs had fatter legs than modern lizards. If dinosaurs had legs in proportion to small lizards the mass to be supported rises as length to the power 3 but the strength of the legs only rises as the area of the cross section, power 2. Therefore the bigger the animal the more enormous the legs must become, which is why a rhino is a very chunky looking version of a pig.

There is a very good article on this in Cooper, Necia Grant; West, Geoffrey B.,

*Particle Physics: A Los Alamos Primer*, ISBN 0521347807## Making tables

edit# Statistics

edit**A reader has identified this chapter as an undeveloped draft or outline.**

You can help to develop the work, or you can ask for assistance in the project room.## Definition of errors

editFor a quantity the error is defined as . Consider a burette which can be read to ±0.05 cm

^{3}where the volume is measured at 50 cm^{3}.- The
*absolute error*is - The
*fractional error*is , - The
*percentage error*is %

## Combination of uncertainties

editIn an experimental situation, values with errors are often combined to give a resultant value. Therefore, it is necessary to understand how to combine the errors at each stage of the calculation.

### Addition or subtraction

editAssuming that and are the errors in measuring and , and that the two variables are combined by addition or subtraction, the uncertainty (absolute error) may be obtained by calculating

which can the be expressed as a relative or percentage error if necessary.

### Multiplication or division

editAssuming that and are the errors in measuring and , and that the two variables are combined by multiplication or division, the fractional error may be obtained by calculating

# Plotting graphs

edit**A reader has identified this chapter as an undeveloped draft or outline.**

You can help to develop the work, or you can ask for assistance in the project room.## The properties of graphs

editThe most basic relationship between two variables and is a straight line, a

*linear relationship*.The variable is the gradient and is a constant which gives the intercept. The equations can be more complex than this including higher powers of , such as

This is called a

*quadratic equation*and it follows a shape called a parabola. High powers of can occur giving*cubic*,*quartic*and*quintic*equations. In general, as the power is increased, the line mapping the variables wiggles more, often cutting the -axis several times.### Practice

editPlot between -3 and +2 in units of 1.

Plot between -4 and +1 in units of 1.

Plot between -5 and +4 in units of 1.

# Complex numbers

edit- ==Introduction to complex numbers==

The equation:

Does not factorise

without complex numbers does not exist. However the number behaves exactly like any other number in algebra without any anomalies, allowing us to solve this problem.

The solutions are .

is an

*imaginary*number. is a*complex*number.Two complex numbers are added by .

Subtraction is obvious: .

Division can be worked out as an exercise. It requires as a common denominator. This is , (difference of two squares), and is .

This means

In practice complex numbers allow one to simplify the mathematics of magnetism and angular momentum as well as completing the number system.

There is an apparent one to one correspondence between the Cartesian plane and the complex numbers, . This is called an Argand diagram. The correspondence is illusory however, because say for example you raise the square root of to a series of ascending powers. Rather than getting larger it goes round and round in circles around the origin. This is not a property of ordinary numbers and is one of the fundamental features of behaviour in the complex plane.

Plot on the same Argand diagram

Solve

(Answers -2 plus or minus 5i, 3/2 plus or minus 2i, i(-1 plus or minus root 2)

2 important equations to be familiar with, Euler's equation:

and de Moivre's theorem:

Euler's equation is obvious from looking at the Maclaurin expansion of .

To find the square root of we use de Moivre's theorem.

so de Moivre's theorem gives

Check this by squaring up to give .

The other root comes from:

de Moivre's theorem can be used to find the

*three*cube roots of unity bywhere can be .

Put , and

This is the difference of two squares so

Similarly any collection of th roots of 1 can be obtained this way.

Another example is to get the expressions for and without expanding .

Remember Pascal's Triangle

1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

*Separating the real and imaginary parts*gives the two expressions. This is a lot easier thanUse the same procedure to get

and .

# Trigonometry

edit- == Trigonometry - the sin and cosine rules ==

In the following trigonometric identities, , and are the lengths of sides in a triangle, opposite the corresponding angles , and .

- The sin rule
- The cosine rule
- The ratio identity

## Trigonometric identities

editRemember this is a consequence of Pythagoras' theorem where the length of the hypotenuse is 1.

The difference of two angles can easily be generated by putting and remembering and . Similarly, the double angle formulae are generated by induction. is a little more complicated but can be generated if you can handle the fractions! The proofs are in many textbooks but as a chemist it is not necessary to know them, only the results.

## Identities and equations

editIdentities and equations look very similar, two things connected by an equals sign. An identity however is a memory aid of a mathematical equivalence and can be proved. An equation represents new information about a situation and can be solved.

For instance,

is an identity. It cannot be solved for ; it is valid for all . However,

is an equation where .

If you try and solve an identity as an equation you will go round and round in circles getting nowhere, but it would be possible to dress up into a very complicated expression which you could mistake for an equation.

## Some observations on triangles

editCheck you are familiar with your elementary geometry. Remember from your GCSE maths the properties of equilateral and iscoceles triangles. If you have an iscoceles triangle you can always dispense with the sin and cosine rules, drop a perpendicular down to the base and use trig directly. Remember that the bisector of a side or an angle to a vertex cuts the triangle in two by area, angle and length. This can be demonstrated by drawing an obtuse triangle and seeing that the areas are .

## The interior angles of a polygon

editRemember that the interior angles of a -sided polygon are (in degrees) or ( ) (in radians)

For benzene there are six

*equilateral*triangles if the centre of the ring is used as a vertex, each of which having an interior angle of 120 degrees. Work out the angles in azulene, (a hydrocarbon with a five and a seven membered ring), assuming all the C-C bond lengths are equal, (this is only approximately true).# Vectors

edit- ==Free Web Based Material from HEFCE==

There is a DVD on vectors at Math Tutor.

## Vectors

editImagine you make a rail journey from Doncaster to Bristol, from where you travel up the West of the country to Manchester. Here you stay a day, travelling the next morning to Glasgow, then across to Edinburgh. At the end of a day's work you return to Doncaster. Mathematically this journey could be represented by vectors, (in 2 dimensions because we are flat earthers on this scale). At the end of the 2nd journey (D-B) + (B-M) you are only a short distance from Doncaster, 50 miles at 9.15 on the clockface. Adding two more vectors, (journeys) takes you to Edinburgh, (about 250 miles at 12.00). Completing the journey leaves you at a zero vector away from Doncaster,

*i.e.*all the vectors in this closed path add to zero.Mathematically we usually use 3 dimensional vectors over the 3 Cartesian axes , and .

It is best always to use the conventional right handed axes even though the other way round is equally valid if used consistently. The wrong handed coordinates can occasionally be found erroneously in published research papers and text books. The memory trick is to think of a sheet of graph paper, is across as usual and up the paper. Positive then comes out of the paper.

A

*unit vector*is a vector*normalised*,*i.e.*multiplied by a constant so that its value is 1. We have the unit vectors in the 3 dimensions:so that

The hat on the i, j, k signifies that it is a

*unit vector*. This is usually omitted.Our geographical analogy allows us to see the meaning of vector addition and subtraction. Vector

**products**are less obvious and there are two definitions the*scalar product*and the*vector product*. These are different kinds of mathematical animal and have very different applications. A scalar product is an area and is therefore an ordinary number, a scalar. This has many useful trigonometrical features.The vector product seems at first to be defined rather strangely but this definition maps onto nature as a very elegant way of describing

*angular momentum*. The structure of Maxwell's Equations is such that this definition simplifies all kinds of mathematical descriptions of atomic / molecular structure and electricity and magnetism.## A summary of vectors

editThe unit vectors in the 3 Cartesian dimensions:

a vector is:

The hat on the i, j, k signifies that it is a

*unit vector*.### Vector magnitude

edit### A constant times a vector

edit### Vector addition

editNotice

### Vector subtraction

editNotice

### Scalar Product

editNotice

Notice that if this reduces to a square.

If A and B have no common non-zero components in , and the value is zero corresponding to

*orthogonality*,*i.e.*they are at right angles. (This can also occur by sign combinations making zero. corresponding to non axis-hugging right angles.)### Vector product

editNotice

The minus sign on the middle term comes from the definition of the determinant, explained in the lecture. Determinants are defined that way so they correspond to right handed rotation. (If you remember our picture of going round the circle, as one coordinate goes up,

*i.e.*more positive, another must go down. Therefore rotation formulae must have both negative and positive terms.) Determinants are related to rotations and the solution of simultaneous equations. The solution of simultaneous equations can be recast in graphical form as a rotation to a unit vector in -dimensional space so therefore the same mathematical structures apply to both space and simultaneous equations.# Matrices and determinants

edit- ==Simultaneous linear equations==

If we have 2 equations of the form we may have a set of

*simultaneous equations*. Suppose two rounds of drinks are bought in a cafe, one round is 4 halves of orange juice and 4 packets of crisps. This comes to 4 pounds 20. The thirstier drinkers at another table buy 4 pints of orange juice and only 1 packet of crisps and this comes to 6 pounds 30. So we have:and

*i.e.*If you plot these equations they will be simultaneously true at and .

Notice that if the two rounds of drinks are 2 pints and 2 packets of crisps and 3 pints and 3 packets of crisps we cannot solve for the prices! This corresponds to two parallel straight lines which never intersect.

If we have the equations:

If these are simultaneously true we can find a

*unique*solution for both and .By subtracting the 2 equations a new equation is created where has disappeared and the system is solved.

Substituting back gives us .

This was especially easy because had the same coefficient in both equations. We can always multiply one equation throughout by a constant to make the coefficients the same.

If the equations were:

and

things would go horribly wrong when you tried to solve them because they are two copies of the same equation and therefore not

*simultaneous*. We will come to this later, but in the meantime notice that 3 times 8 = 4 times 6. If our equations were:we can still solve them but would require a lot of algebra to reduce it to three (2x2) problems which we know we can solve. This leads on to the subject of matrices and determinants.

Simultaneous equations have applications throughout the physical sciences and range in size from (2x2)s to sets of equations over 1 million by 1 million.

## Practice simultaneous equations

editSolve:

and

Notice that you can solve:

because it breaks down into a (2x2) and is not truly a (3x3). (In the case of the benzene molecular orbitals, which are (6x6), this same scheme applies. It becomes two direct solutions and two (2x2) problems which can be solved as above.)

## Matrices

editThe multiplication of matrices has been explained in the lecture.

but cannot exist. To be multiplied two matrices must have the 1st matrix with the same number of elements in a row as the 2nd matrix has elements in its columns.

where the s are the elements of .

Look at our picture of and as represented by a unit vector in a circle. The rotation of the unit vector about the -axis can be represented by the following mathematical construct.

In two dimensions we will rotate the vector at 45 degrees between and :

This is if we rotate by +45 degrees. For and . So the rotation flips over to give . The minus sign is necessary for the correct mathematics of rotation and is in the lower left element to give a right handed sense to the rotational sign convention.

As discussed earlier the solving of simultaneous equations is equivalent in some deeper sense to rotation in -dimensions.

## Matrix multiply practice

editi) Multiply the following (2x2) matrices.

=

ii) Multiply the following (3x3) matrices.

You will notice that this gives a

*unit matrix*as its product.The first matrix is the

*inverse*of the 2nd. Computers use the inverse of a matrix to solve simultaneous equations.

If we have

In matrix form this is....In terms of work this is equivalent to the elimination method you have already employed for small equations but can be performed by computers for simultaneous equations.

(Examples of large systems of equations are the fitting of reference data to 200 references molecules, dimension 200, or the calculation of the quantum mechanical gradient of the energy where there is an equation for every way of exciting 1 electron from an occupied orbital to an excited, (called

*virtual*, orbital, (typically equations.)## Finding the inverse

editHow do you find the inverse... You use Maple or Matlab on your PC but if the matrix is small you can use the formula...

Here Adj A is the adjoint matrix, the transposed matrix of cofactors. These strange objects are best described by example.....

This determinant is equal to: 1 ( 1 x 1 - 1 x (-1)) - (-1) ( 2 x 1 - 1 x 3) + 2 ( 2 x (-1) - ( 1 x 3) each of these terms is called a

*cofactor*.

This thing gives the sign alternation in a form mathematicians like even though it is incomprehensible.

Use the determinant

to solve the simultaneous equations on page 47 by the matrix inverse method. The matrix corresponding to the equations on p47.2 is:

1 -1 2 6 2 1 1 = 3 3 -1 1 6 The cofactors are 2 1 -5 -1 -5 -2 -3 3 3 You may find these 9 copies of the matrix useful for striking out rows and columns to form this inverse.... 1 -1 2 1 -1 2 1 -1 2 2 1 1 2 1 1 2 1 1 3 -1 1 3 -1 1 3 -1 1 1 -1 2 1 -1 2 1 -1 2 2 1 1 2 1 1 2 1 1 3 -1 1 3 -1 1 3 -1 1 1 -1 2 1 -1 2 1 -1 2 2 1 1 2 1 1 2 1 1 3 -1 1 3 -1 1 3 -1 1 These are the little determinants with the -1 to the (n-1) factors and the value of the determinant is -9. The transposed matrix of cofactors is 2 -1 -3 1 -5 3 -5 -2 3 So the inverse is 2 -1 -3 -1/9 X 1 -5 3 -5 -2 3 Giving a solution 2 -1 -3 6 1 -1/9 X 1 -5 3 X 3 = -1 -5 -2 3 6 2

This takes a long time to get all the signs right. Elimination by subtracting equations is MUCH easier. However as the computer cannot make sign mistakes it is not a problem when done by computer program.

The following determinant corresponds to an equation which is repeated three times giving an unsolvable set of simultaneous equations.

Matrix multiplication is not necessarily commutative, which in English means does not equal all the time. Multiplication may not even be possible in the case of rectangular rather than square matrices.

I will put a list of the properties and definitions of matrices in an appendix for reference through the later years of the course.

## Determinants and the Eigenvalue problem

editIn 2nd year quantum chemistry you will come across this object:

You divide by and set to equal to get:

Expand this out and factorise it into two quadratic equations to give:

which can be solved using

### Simultaneous equations as linear algebra

editThe above determinant is a special case of simultaneous equations which occurs all the time in chemistry, physics and engineering which looks like this:

This equation in matrix form is and the solution is .

This is a

*polynomial equation*like the quartic above. As you know polynomial equations have as many solutions as the highest power of*i.e.*in this case . These solutions can be*degenerate*for example the orbitals in benzene are a degenerate pair because of the factorisation of the polynomial from the 6 Carbon-pz orbitals. In the 2nd year you may do a lab exercise where you make the benzene determinant and see that the polynomial isfrom which the 6 solutions and the orbital picture are immediately obvious.

The use of matrix equations to solve arbitrarily large problems leads to a field of mathematics called

*linear algebra*.## Matrices with complex numbers in them

editWork out the quadratic equation from the 3 determinants

They are all the same! This exemplifies a deeper property of matrices which we will ignore for now other than to say that complex numbers allow you to calculate the same thing in different ways as well as being the

*only*neat way to formulate some problems.# Differentiation

edit- ==Free web-based material from HEFCE==

There is a DVD on differentiation at Math Tutor.

## The basic polynomial

editThe most basic kind of differentiation is:

There are two simple rules:

- The derivative of a function times a constant is just the same constant times the derivative.
- The derivative of a sum of functions is just the sum of the two derivatives.

To get higher derivatives such as the

*second derivative*keep applying the same rules.One of the big uses of differentiation is to find the stationary points of functions, the maxima and minima. If the function is smooth, (unlike a saw-tooth), these are easily located by solving equations where the first derivative is zero.

## The chain rule

editThis is best illustrated by example: find given

Let and .

Now and

So using the chain rule we have

## Differentiating a product

editNotice when differentiating a product one generates two

*terms*. (Terms are mathematical expression connected by a plus or minus.) An important point is that terms which represent physical quantities must have the same units and dimensions or must be pure dimensionless numbers. You cannot add 3 oranges to 2 pears to get 5 orangopears. Integration by parts also generates an extra term each time it is applied.## Differentiating a quotient

editYou use this to differentiate .

### Problems

editDifferentiate with respect to

Notice we have .

Evaluate the inner brackets first.

Evaluate

a, b and c are constants. Differentiate with respect to .

### Answers

edit### Harder differentiation problems

editDifferentiate with respect to :

Differentiate with respect to

Differentiate with respect to

Evaluate

## Using differentiation to check turning points

editis the tangent or gradient. At a minimum is zero. This is also true at a maximum or an inflection point. The

*second gradient*gives us the nature of the point. If is positive the turning point is a minimum and if it is negative a maximum. Most of the time we are interested in minima except in*transition state theory*.If the equation of is plotted, is is possible to see that at there is a third kind of point, an inflection point, where both and are zero.

Plot between -4 and +3, in units of 1. (It will speed things up if you factorise it first. Then you will see there are 3 places where so you only need calculate 5 points.) By factorising you can see that this equation has 3 roots. Find the 2 turning points. (Differentiate once and find the roots of the quadratic equation using . This gives the position of the 2 turning points either side of zero. As the equation is only in it has 3 roots and 2 maxima / minima at the most therefore we have solved everything. Differentiate your quadratic again to get . Notice that the turning point to the left of zero is a maximum

*i.e.*and the other is a minimum*i.e.*.What is the solution and the turning point of .

Solve , by factorisation.

(The 3 roots are -3,0 and +2.

Solutions are and ,

*i.e.*-1.7863 and 1.1196.There are 3 coincident solutions at , , at 0 so this is an

*inflection point*.The roots are 0, 1 and -1.

# Integration

edit- ==Free Web Based Material from HEFCE==

There is a DVD on integration at Math Tutor.

### The basic polynomial

editThis works fine for all powers except -1, for instance the integral of

is just

-1 is clearly going to be a special case because it involves an infinity when and goes to a steep spike as gets small. As you have learned earlier this integral is the natural logarithm of and the infinity exists because the

*log*of zero is minus infinity and that of negative numbers is undefined.### The integration and differentiation of positive and negative powers

edit>>>>>>>>>>>>>>>>>>>>>>Differentiation 1/3 x*x*x x*x 2x 2 0 0 0 0 1/3 x*x*x x*x 2x 2 ? ? ? ? Integration<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< I(x) H(x) G(x) F(x) ln(x) 1/x -1/(x*x)

Here I, H, G and F are more complicated functions involving .You will be able to work them out easily when you have done more integration. The thing to notice is that the calculus of negative and positive powers is not symmetrical, essentially caused by the

*pole*or*singularity*for at .## Logarithms

editLogarithms were invented by Napier, a Scottish Laird, in the 17th-century. He made many inventions but his most enduring came from the necessity of doing the many long divisions and trigonometric calculations used in astronomy. The Royal Navy in later years devoted great time and expense to developing logarithm technology for the purposes of navigation, leading to the commissioning of the first mechanical stored program computer, which was theoretically sound but could not be made by Charles Babbage between 1833 and 1871.

The heart of the system is the observation of the properties of powers:

This means that if we have the

*inverse*function of we can change a long division into a subtraction by looking up the exponents in a set of tables. Before the advent of calculators this was the way many calculations were done.Napier initially used logs to the base for his calculations, but after a year or so he was visited by Briggs who suggested it would be more practical to use base 10. However base is necessary for the purposes of calculus and thermodynamics.

## Integrating 1/x

editThis is true because is our function which reduces or grows at the rate of its own quantity.

This is our definition of a logarithm.

therefore

therefore

## Integrating 1/x like things

editJust as

so

therefore by the

*chain rule*therefore

Examples of this are:

or

and

As the integral of is so the differential of is so

is just a constant so so

This can also be done by the chain rule

What is interesting here is that the 5 has disappeared completely. The gradient of the log function is unaffected by a multiplier! This is a fundamental property of logs.

## Some observations on infinity

editObviously is .

nd are undefined but sometimes a large number over a large number can have defined values. An example is the of 90 degrees, which you will remember has a large opposite over a large hypotenuse but in the limit of an infinitesimally thin triangle they become equal. Therefore the is 1.

## Definite integrals (limits)

editRemember how we do a definite integral

where is the indefinite integral of .

Here is an example where limits are used to calculate the 3 areas cut out by a quartic equation:

.

We see that is a solution so we can do a polynomial division:

x3 -x2 -2x ------------------- x-1 ) x4 -2x3 -x2 +2x x4 -x3 -------- -x3 -x2 -x3 +x2 ------- -2x2 +2x -2x2 +2x -------- 0

So the equation is which factorises to

.

## Integration by substitution

editwhere .

## Simple integration of trigonometric and exponential Functions

edit### Answers

edit*i.e.*the integral of .## Integration by parts

editThis is done in many textbooks and Wikipedia. Their notation might be different to the one used here, which hopefully is the most clear. You derive the expression by taking the product rule and integrating it. You then make one of the into a product itself to produce the expression.

(all integration with respect to . Remember by

( gets differentiated.)

The important thing is that you have to integrate one expression of the product and differentiate the other. In the basic scheme you integrate the most complicated expression and differentiate the simplest. Each time you do this you generate a new

*term*but the function being differentiated goes to zero and the integral is solved. The expression which goes to zero is .The other common scheme is where the parts formula generates the expression you want on the right of the equals and there are no other integral signs. Then you can rearrange the equation and the integral is solved. This is obviously very useful for trig functions where

*ad infinitum*.also generates itself and is susceptible to the same treatment.

We now have our required integral on both sides of the equation so

## Integration Problems

editIntegrate the following by parts with respect to .

Actually this one can be done quite elegantly by parts, to give a two term expression. Work this one out. Expanding the original integrand by Pascal's Triangle gives:

2 3 4 5 6 7 8 x + 7 x + 21 x + 35 x + 35 x + 21 x + 7 x + x

The two term integral expands to2 3 4 5 6 7 8 9 1/2 x + 7/3 x + 21/4 x + 7 x + 35/6 x + 3 x + 7/8 x + 1/9 x - 1/72

So one can see it is correct on a term by term basis.

If you integrate you will have to apply

*parts*7 times to get to become 1 thereby generating 8 terms:7 6 5 4 3 -x cos(x) + 7 x sin(x) + 42 x cos(x) - 210 x sin(x) - 840 x cos(x) + 2 2520 x sin(x) + 5040 x cos(x) - 5040 sin(x) + c (Output from Maple.)

Though it looks nasty there is quite a pattern to this, 7, 7x6,7x6x5 ------7! and sin, cos, -sin, -cos, sin, cos

*etc*so it can easily be done by hand.## Differential equations

editFirst order differential equations are covered in many textbooks. They are solved by integration. (First order equations have , second order equations have and .)

The arbitrary constant means another piece of information is needed for complete solution, as with the Newton's Law of Cooling and Half Life examples.

Provided all the s can be got to one side and the s to another the equation is separable.

This is the

*general solution*.Typical examples are:

by definition of logs.

(1)

(2)

(3)

This corresponds to:

The Schrödinger equation is a 2nd order differential equation

*e.g.*for the particle in a boxIt has taken many decades of work to produce computationally efficient solutions of this equation for polyatomic molecules. Essentially one expands in coefficients of the

*atomic orbitals*. Then integrates to make a differential equation a set of numbers, integrals, in a*matrix*. Matrix algebra then finishes the job and finds a solution by solving the resultant simultaneous equations.## The calculus of trigonometric functions

editThere are many different ways of expressing the same thing in trig functions and very often successful integration depends on recognising a trig identity.

but could also be

(each with an integration constant!).

When applying calculus to these functions it is necessary to spot which is the simplest form for the current manipulation. For integration it often contains a product of a function with its derivative like where integration by substitution is possible.

Where a derivative can be spotted on the numerator and its integral below we will get a function. This is how we integrate .

We can see this function goes to infinity at as it should do.

## Integration by rearrangement

editTake for example:

Here there is no function producted in with the powers so we cannot use substitution. However there are the two trig identities

and

Using these we have

so we have two simple terms which we can integrate.

## The Maclaurin series

editWe begin by making the assumption that a function can be approximated by an infinite power series in :

By differentiating and setting one gets

Sin, cos and can be expressed by this series approximation

Notice also works for negative .

When differentiated or integrated generates itself!

When differentiated generates .

By using series we can convert a complex function into a polynomial, and can use for small .

In actual fact the kind of approximation used inside computer programs is more like:

These have greater range but are much harder to develop and a bit fiddly on the calculator or to estimate by raw brain power.

We cannot expand this way because is

. However can be expanded.

Work out the series for .

### Factorials

editThe factorials you have seen in series come from repeated differentiation. also has a statistical meaning as it is the number of unique ways you can arrange objects.

- The