What is a line? In Euclidean Geometry we usually don’t define a line and instead call it a primitive object (*the properties of lines are then determined by the axioms which refer to them*). If instead points and line *segments* – defined by pairs of points –* * are taken as the primitive objects, the following might define lines:

Geometric Definition CandidateA

line,, is a set of points with the property that for each pair of points in the line, ,.

In terms of a picture this just says that when you have a line, that if you take two points *in *the line (the language *in *comes from set theory), that the line segment is a subset of the line:

*Why is this *objectively *not a good definition of a line.*

Once we move into Cartesian\Coordinate Geometry we can perhaps do a similar trick. We can use line segments, and their lengths to define slope, (slope = rise over run) and then define a line as follows:

Algebraic Definition CandidateA

line, , is a set of points such that for all pairs ofdistinctpoints , the slope is a constant.

This means that if you take two pairs of distinct points in a line , and then calculate the slopes between them, you get the same answer, and therefore it makes sense to talk about *the *slope of a line, .

This definition, however, has exactly the same problem as the previous. The definition we use isn’t too important but I do want to use a definition that considers the line a *set of points.*

We can use such a definition to derive the equation of a line ‘formula’ for a line of slope containing a point .

Suppose first of all that we have an axis and a point in the line. What does it take for a second point to be in the line?

The point is on the line if and only if the slope between and is equal to :

.

For any line at all this can be rewritten as , with the slope and the -coordinate when : i.e. the -intercept:

*Parallel* and *perpendicular *are then *relations* on the set of lines . This gives yet another definition of a line (that does not, by the way, include vertical lines. Again, we can include them with small tweaks to this presentation).

Algebraic DefinitionA

line, , of slope and -intercept , is the set of solutions (in the plane) to the equation.

A point is on the line if and only if it satisfies the equation:

.

It is probably more natural to come at these concepts first from a geometric angle (pun not intended). There are various definitions. For example, The Penguin Dictionary of Mathematics gives the following description:

Describing lines, curves, planes, or surfaces that are always equidistant, and that will never meet no matter how far they are produced. Parallel lines and curves must both lie in the same plane.

Whatever the definition, we know a pair of parallel lines when we see them.

One implication of this definition is that a line is not parallel to itself. This illustrates a feature of mathematics that is perhaps not appreciated at the primary and secondary levels, namely that

It’s sometimes hard for people learning mathematics, who naturally feel that mathematics is an objective discipline, to hear that many things are actually a matter of convention.

We have seen a similar thing above: we usually take lines themselves as primitive objects and define line segments in terms of lines. It seems to be possible to define lines in terms of line segments (although the two definitions above do not achieve this).

There is a choice here and the choice made dictates the detail of the sequel but not the essence… it is going to be slightly easier for me to present things if I say that a line is *not *parallel to itself (although if we do say this we get that ‘parallel’ is an equivalence relation). There are other choices but here is the definition that I will use is as follows.

Geometric DefinitionA line isparallelto a line , written , if , the empty set.

So rather than explicitly including the equi-distance I am just going to say that a pair of lines are parallel if they do not intersect. I am almost completely sure that equi-distance is a consequence of the above definition (perhaps with an additional axiom from Euclidean Geometry).

What we will show now is that parallel lines have the same slope.

Suppose that and are two parallel lines with equations:

and .

By the Geometric Definition, these lines do not intersect. That means there is no point that satisfies both equations, so no simultaneous solution to the equations. This means there is no solution to:

(*)

.

This clearly is a solution *unless *there is a division by zero, that is unless .

That is, parallel lines have the same slope. Note that (*) has a solution if but this is not allowed by the definition of parallel that I am using.

This suggests the following *algebraic* definition for parallel.

Algebraic DefinitionA line is

parallelto adifferentline , written ,if the lines have the same slope.

If we think of slope as measuring steepness/direction, this definition is quite natural.

If we take this as the definition then we can derive the geometric definition. Suppose that and are parallel. Therefore their equations are

and ,

with .

To find the intersection we solve the simultaneous equations. Doing this leads to which is false (the two lines must be different according to the definition). Therefore there is no intersection.

We have shown (with the restriction of lines to non-vertical lines) that

Geometric Definition Algebraic Definition, and

Algebraic Definition Geometric Definition

This implies that the definitions are equivalent. This post introduces the idea of Duality: in this case between Geometry and Algebra. The power of this duality is that sometimes it is easier to consider ideas/questions in the geometry picture, and sometimes it is easier to consider the algebra picture. The discussion here shows that when convenient we can think of parallel lines as being lines that do not intersect (geometry (or maybe even a set theory picture?)), and when convenient we can think of them as (distinct) lines that have the same slope; and that these are equivalent pictures.

Now we try the same kind of argument with perpendicularity.

Again, we should know perpendicular lines when we see them:

Lines are perpendicular when the *angle* between them is a right angle, . We don’t have to write down a good definition of angle: let us just say that an *ordered* triple defines an angle, with , the ‘common’ point:

is perpendicular to because .

As there are four angles involved, with two of them equal, and they add up to , we can define *the *angle between two lines as the smallest of the four and it is automatically less than or equal to .

This time, the candidate for an algebraic definition, using slopes, does not seem natural (although it is possible to use *vectors *and the dot product to generate a natural algebraic definition (that has the great property of being generalisable to more abstract settings)). Therefore we will simply write down the Geometric Definition and then prove an Algebraic *Condition *that holds for perpendicular lines. Again we restrict to non-vertical lines.

Geometric DefinitionLines and are

perpendicular, written , ifthe angle betweenand is .

Now consider perpendicular lines and with slopes and :

Without affecting the slopes of and we can consider a new -axis that has the same scale as the old but has as origin at :

So we can assume if we want that the perpendicular lines both go through the origin. Now go to and drop perpendiculars up and down to and as shown:

Now if we call the point by , note that and are both right-angled triangles and so Pythagoras Theorem applies:

and .

Calculate the length of using slope.

.

Similarly .

Now consider the right-angled triangle with hypotenuse and other sides such that and .

Apply Pythagoras Theorem to to show that:

AlgebraicConditionLines and are

perpendicular, written , if .

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If you would like to submit anonymous feedback on this module/lecturer, you may do so here. This link will be open until Friday May 11 2018.

On Monday we finished the module by looking at triple integrals.

The Wednesday 09:00 lecture will be a tutorial. In this class and your usual tutorial we will look at the P.182, P. 192, P. 163, & P.116 exercises. If these are completed you will be recommended to revise either by trying Chapter 1 & 2 exercises or perhaps by looking at the Summer 2017 paper.

As next Monday is a bank holiday, we will begin the Summer 2017 Paper (in your notes) revision on Thursday.

In the Wednesday 09:00 lecture we will continue working on the Summer 2017 Paper and hopefully finish it before the end of the Thursday 10:00 lecture.

If we finish the Summer 2017 paper early, any extra time (probably just Thursday but maybe Wednesday if we go fast) will be dedicated to one-to-one help.

Wednesday’s tutorial will go ahead as normal with one-to-one help.

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

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If you would like to submit anonymous feedback on this module/lecturer, you may do so here. This link will be open until Friday May 11 2018.

Assignment 2 has been corrected and your results emailed to you.

We spent three lectures looking at double integrals, in particular their application to second moments of area. We set of integration over a cylinder by looking at polar coordinates.

In the Wednesday tutorial we worked on the p. 163 (primarily) and p. 182 exercises.

On Monday we will finish the module by looking at triple integrals.

We will therefore have three tutorials where we will look at the P.182, P. 192, P. 163, & P.116 exercises. If these are completed you will be recommended to revise either by trying Chapter 1 & 2 exercises or perhaps by looking at the Summer 2017 paper.

We will go through last year’s exam on the board and then I will answer your questions if there are any. If there are none I will help one-to-one.

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

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If you would like to submit anonymous feedback on this module/lecturer, you may do so here. This link will be open until Friday May 11 2018.

We looked at the normal distribution.

In Maple looked at Binomial and Poisson random variables.

The Maple Test should take no more than one hour but I am giving ye extra time. For various reasons, I have decided to schedule next week’s class as:

- 19:05 – 20:25: Sampling and Control Charts
- 20:25 – 20:45: Break
- 20:45 – 22:00: Maple Test

The Maple Test will be open book. You have a sample Maple Test (this is also in the notes) with solutions (*the first with(Statistics) should be with(LinearAlgebra)). The Maple Test will not include anything from Chapter 2 (Lab 4).

We will speak about sampling in more detail and also introduce control charts.

We will hold a review class on Wednesday 9 May in the usual room. First off, the *layout* of your exam is the same as Autumn 2016 (in the back of your notes): do question one worth 50/100 and two out of questions two, three, four; each worth 25/100.

I will field any questions ye might have at this time and if there are no questions we will do this exam paper. The best possible thing for your study is to do this exam paper and then on Wednesday see how you got on.

If you have missed a lab you have two options: either download Maple onto your own machine (instructions may be found here) or come into CIT at another time to use Maple.

Go through the missed lab on your own, doing *all* the exercises in Maple. Save the worksheet and email it to me.

The deadline for Maple Catch up is Friday May 11 2018.

Questions you can do include:

**After Week 11:**P. 124, Q. 1-10 (this is loads: more is Q. 11-21)**After Week 10:**P. 102, Q. 1-4; P. 107, Q. 1-7; P. 111, Q. 1-12 (this is loads: more is Q. 13-16); P. 115, Q. 1-15**After Week 9:**P. 92, Q. 1-10 (not too important); P. 96, Q. 1-6 (this is loads: more is Q. 7-13)**After Week 8*:**P. 89, Q. 1-3**After Week 7*:**P. 89, Q. 1 (a), (b); Q. 2 (b); Q. 3 (b)**After Week 6*:**P. 74, Q. 1-4; P. 77, Q. 1-3**After Week 5*:**P.44, Q. 1-3, Q. 4-5 more abstract. P.47, Q. 1-3, Q. 4 more abstract. P.56, Q. 1-3, Q. 4 more abstract. P.69, Q. 9 is an important question. A version might be

Use only

determinantsto determine if the following homogeneous system of linear equations has non-zero solutions:

**After Week 4:**P. 41, Q. 1-4**After Week 3:**P. 28, Q. 1-5, 6-9 have answers with Q. 7 a harder question. P. 34 exercises.**After Week 2:**P. 18 Q. 2**After Week 1:**P. 18 Q. 1, 3 – 6. Harder questions are 7 and 8. For those who do not yet have the manual, see here.

I am not suggesting you should do *all *of these. It is recommended by the module descriptor that you do two hours of independent and directed learning every week but of course this isn’t feasible for everyone.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

]]>We did a lot of probability — reliability block diagrams, the binomial distribution, the Poisson distribution.

Those who missed the class I recorded some of it here.

Ironically I recorded the same lectures in 2016 as you can see here.

We will look at the normal distribution and talk about sampling.

In Maple we will look at Binomial and Poisson random variables.

We will speak about sampling in more detail and also introduce control charts.

The Maple Test will be open book. You have a sample Maple Test with solutions (*the first with(Statistics) should be with(LinearAlgebra)). The Maple Test will not include anything from Chapter 2.

We will hold a review class on Wednesday 9 May in the usual room. First off, the *layout* of your exam is the same as Autumn 2016 (in the back of your notes): do question one worth 50/100 and two out of questions two, three, four; each worth 25/100.

I will field any questions ye might have at this time and if there are no questions we will do this exam paper. The best possible thing for your study is to do this exam paper and then on Wednesday see how you got on.

If you have missed a lab you have two options: either download Maple onto your own machine (instructions may be found here) or come into CIT at another time to use Maple.

Go through the missed lab on your own, doing *all* the exercises in Maple. Save the worksheet and email it to me.

Questions you can do include:

**After Week 10:**P. 102, Q. 1-4; P. 107, Q. 1-7; P. 111, Q. 1-12 (this is loads: more is Q. 13-16); P. 115, Q. 1-15**After Week 9:**P. 92, Q. 1-10 (not too important); P. 96, Q. 1-6 (this is loads: more is Q. 7-13)**After Week 8*:**P. 89, Q. 1-3**After Week 7*:**P. 89, Q. 1 (a), (b); Q. 2 (b); Q. 3 (b)**After Week 6*:**P. 74, Q. 1-4; P. 77, Q. 1-3**After Week 5*:**P.44, Q. 1-3, Q. 4-5 more abstract. P.47, Q. 1-3, Q. 4 more abstract. P.56, Q. 1-3, Q. 4 more abstract. P.69, Q. 9 is an important question. A version might be

Use only

determinantsto determine if the following homogeneous system of linear equations has non-zero solutions:

**After Week 4:**P. 41, Q. 1-4**After Week 3:**P. 28, Q. 1-5, 6-9 have answers with Q. 7 a harder question. P. 34 exercises.**After Week 2:**P. 18 Q. 2**After Week 1:**P. 18 Q. 1, 3 – 6. Harder questions are 7 and 8. For those who do not yet have the manual, see here.

I am not suggesting you should do *all *of these. It is recommended by the module descriptor that you do two hours of independent and directed learning every week but of course this isn’t feasible for everyone.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

]]>Assignment 2 has a hand-in date of 17:00 23 April: the Monday of Week 11. Assignment 2 is in the manual, P. 149..

The Monday lecture was another tutorial on the P.116 exercises. In the Wednesday lectures we worked on systems of differential equations. In the Thursday lecture we worked on systems of differential equations exercises.

In the Wednesday tutorial we continued with the full Laplace Transform questions.

In the Monday and Wednesday lectures we will make a start on the final chapter by looking at double integrals.

In the Wednesday tutorial we will work on the p. 163 and p. 182 exercises. This work will continue on Thursday.

On Monday and Wednesday we will finish looking at double integrals and then triple integrals.

In the Wednesday tutorial we will work on the p.182, p.192, and p.186 exercises. This work will continue on Thursday.

We will go through last year’s exam on the board and then I will answer your questions if there are any. If there are none I will help one-to-one.

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

]]>

We looked at finite differences for the Heat Equation. This completes the examinable written material.

In VBA we implemented same.

In the 09:00 class we will have a revision session, geared towards the 20% VBA Assessment 2. This will look at the *20% VBA Assessment 2 Tutorial* Sheet It might therefore be a good idea to go through this before next week.

In the 12:00 class we will have a revision session, geared towards the 40% Written Assessment 2. This will look at the *40**% Written Assessment 2 Tutorial* Sheet It might therefore be a good idea to go through this before next week.

Formulae will be provided in the VBA 2 Assessment.

To understand how your student numbers generate constants (see below) see this VBA Test 2 from last year (do *not *read this as a sample – it included e.g. the Heat Equation which you will not be examined on (in VBA) and the Laplace’s Equation might be slightly simpler than what ye will have).

See last week’s Weekly Summary for the Format of the VBA Assessment 2.

The 40% Written Test is to be split in two.

The first part of the Written Test is the Theory Element (worth 20%). It will take place in B242 at 09:00, Tuesday 1 May.

From the *40**% Written Assessment 2 Tutorial* Sheet, you will receive

- one question from
*Runge-Kutta Exercises*[10%] - 3/4 questions from
*Other Theory Exercises*[10%]

This Theory Element is designed to take 30 minutes. You will be allowed up to 55 minutes.

The second part of the Written Test is the Calculations Element (worth 20%). It will take place during your Week 12 VBA slot. Like the *40**% Written Assessment 2 Tutorial* Sheet, you will receive

- A damped harmonic oscillator, with for all groups but will be different for groups A, B, and C [8%]
- A Heat Flux Density Vector Question, with different temperature distributions for groups A, B, and C. [4%]
- A Heat Equation Question. With different values for groups A, B, and C [8%]

This Calculation Element is designed to take 45 minutes. You will be allowed up to an hour and 55 minutes.

There will be no 12:00 class on Tuesday 1 May.

Study should consist of

- doing exercises from the notes
- completing VBA exercises

Assignment 2 has a hand-in date of 17:00 23 April: the Monday of Week 11. Assignment 2 is in the manual, P. 149.

If you were absent today and want to view your submission next week please email me.

We said a few things about damped harmonic oscillators on Monday and the rest of the week was spent working on the p.116 exercises.

The Monday lecture will be another tutorial on the P.116 exercises. In the Wednesday and Thursday lectures we will work on systems of differential equations.

In the Wednesday tutorial we will continue with the p.116 exercises.

In the Monday and Wednesday lectures we will make a start on the final chapter by looking at double integrals.

In the Wednesday tutorial we will work on the p. 163 and p. 182 exercises. This work will continue on Thursday.

On Monday and Wednesday we will finish looking at double integrals and then triple integrals.

In the Wednesday tutorial we will work on the p.182, p.192, and p.186 exercises. This work will continue on Thursday.

We will go through last year’s exam on the board and then I will answer your questions if there are any. If there are none I will help one-to-one.

Please feel free to ask me questions about the exercises via email or even better on this webpage.

]]>

We made a good start on probability, talking about random variables, independence, mutual exclusivity, conditional probability, and tree diagrams.

You can read about the child paradox here.

We will have a lot of probability to do — reliability block diagrams, the binomial distribution, the Poisson distribution.

You have a sample Maple Test with solutions (*the first with(Statistics) should be with(LinearAlgebra)). The Maple Test will not include anything from Chapter 2.

We do not have enough probability done to have a Maple Lab until…

We will look at the normal distribution and talk about sampling.

We will speak about sampling in more detail and also introduce control charts.

The Maple Test will be open book and you will have already received a sample test with solutions.

We will hold a review class on Wednesday 9 May in the usual room. First off, the *layout* of your exam is the same as Autumn 2016 (in the back of your notes): do question one worth 50/100 and two out of questions two, three, four; each worth 25/100.

I will field any questions ye might have at this time and if there are no questions we will do this exam paper. The best possible thing for your study is to do this exam paper and then on Wednesday see how you got on.

If you have missed a lab you have two options: either download Maple onto your own machine (instructions may be found here) or come into CIT at another time to use Maple.

Go through the missed lab on your own, doing *all* the exercises in Maple. Save the worksheet and email it to me.

Questions you can do include:

**After Week 9:**P. 92, Q. 1-10 (not too important); P. 96, Q. 1-6 (this is loads: more is Q. 7-13)**After Week 8*:**P. 89, Q. 1-3**After Week 7*:**P. 89, Q. 1 (a), (b); Q. 2 (b); Q. 3 (b)**After Week 6*:**P. 74, Q. 1-4; P. 77, Q. 1-3**After Week 5*:**P.44, Q. 1-3, Q. 4-5 more abstract. P.47, Q. 1-3, Q. 4 more abstract. P.56, Q. 1-3, Q. 4 more abstract. P.69, Q. 9 is an important question. A version might be

Use only

determinantsto determine if the following homogeneous system of linear equations has non-zero solutions:

**After Week 4:**P. 41, Q. 1-4**After Week 3:**P. 28, Q. 1-5, 6-9 have answers with Q. 7 a harder question. P. 34 exercises.**After Week 2:**P. 18 Q. 2**After Week 1:**P. 18 Q. 1, 3 – 6. Harder questions are 7 and 8. For those who do not yet have the manual, see here.

I am not suggesting you should do *all *of these. It is recommended by the module descriptor that you do two hours of independent and directed learning every week but of course this isn’t feasible for everyone.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

]]>