Category Archive

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Test 2 Results

Hopefully this week.

Schedule — Recently updated

Thursday 6 December 20:00 – 22:00: Finish double integrals and start Triple integrals

Tuesday 11 December 18:00 – 22:00: Finish off multiple integration; Exam Format Review Lecture; Tutorial; etc. Will make a plan on Friday.

Integral

In class we found the anti-derivative by using the substitution . Here we present an alternative method that manipulates the integrand instead.

Read the rest of this entry »

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Test 2

The second test is now on Thursday 29 November at 6.10 pm sharp (please email me if this is an issue).

Please find a sample.

Schedule — Recently updated

Tuesday 27 November:  Finish off line integrals including Green’s Theorem — a theorem that will tell us when an integral over a closed curve is going to be zero. Tutorial for Test 2

Thursday 29 November: Test 2. Start double integrals.

Thursday 6 December: Finish double integrals and Triple integrals

Tuesday 11 December: Possibly finish off multiple integration; Exam Format Review Lecture; Tutorial; etc. Will make a plan in next two weeks.

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Test 2

The second test is on Tuesday 27 November at 8.20 pm sharp.

I will give ye a sample on Thursday 22 November.

Schedule

Tuesday 20 November: Finish off Chapter 3 and start looking at Multiple Integration (line integrals).

Thursday 22 November: Tutorial for Test 2

Tuesday 27 November: Test 2. Start double integrals.

Tuesday 4 December: Tutorial for line integrals & double integrals

Thursday 6 December: Finish double integrals and Triple integrals

Wednesday 12 December: Possibly finish off multiple integration; Exam Format Review Lecture; Tutorial.

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Lectures

We have finally concluded our study of Laplace methods. Next week we look at Lagrange Interpolation and the Least Squares Method.

Winter Exam Format

I have changed the format of the exam somewhat. It is answer four out of five:

1. Multiple Topics Question

(a) Interpolation (b) Linear Algebra (c) Laplace Methods

2. Interpolation

Finite Difference Methods, Least Squares, Lagrangian Interpolation

3. Linear Algebra

Gaussian Elimination with and without partial pivoting, Cramer’s Rule, Jacobi’s Method, Gauss-Siedel Method

4. Laplace Methods

Three differential equations

5.  Multiple Integration

line integrals, double integrals with applications including second moment of area, triple integrals.

Test 2: Possible Notice

I want to hold Test 2 on Thursday 30 November in Week 11. If ye think this is a problem or is clashing with an established test or deadline please get back onto me.

Test 1 Results

I have them: please email me if you didn’t get yours during Thursday’s class.

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Test 1 Results

I have them: please email me if you didn’t get yours during Thursday’s class.

Lectures

We will spend another session on Laplace Methods and then onto pastures new.

Tutorial

We have a very important tutorial on Tuesday. Lots of work to do:

• Laplace Transform of differential equations — morryah question four from the test. If you don’t get this grab me and make me explain it to you.
• Partial Fractions — once we’ve done the above we will have to do a partial fraction expansion (most of the time) in order to put the transformed solution/function into a form we can deal with.
• Inverse Laplace Transform — Well we’ll try and get some differential equations done and see what comes up.

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Test 1 Results

Pending…

Next Week

We will continue our work on the Inverse Laplace Transform and finally solve some differential equations. This is the metaphor that explains the use of Laplace Methods in the solution of same:

Suppose that you come across a poem written in English of whose meaning you don’t understand. However suppose that you know a French-speaking gentleman who is a master of interpreting poems. So you translate the poem into French and send it to the French gentleman. The French gentleman writes a perfectly good interpretation of the poem in French and sends this back to you where you translate it back into English and you have the meaning of the poem.

1. Poem in English = Differential Equation
2. Interpretation in English = Solution of Differential Equation
3. Translate into French = Take Laplace Transform
4. Poem in French (better interpreter) = Algebraic Equation (easier to solve)
5. Interpretation in French = Laplace Transform of Solution of Differential Equation
6. Translate back into English = Inverse Laplace Transform

More Explanations

Somebody told me the beauty of learning about maths off a video is that you can pause the lecturer. I found these online and I have to say I like these lectures a lot… not many examples but we do them in class even if you seem to think that I don’t!

http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/

We would have covered lectures 11 to 14 last year.

Lectures 19 & 20 are relevant for us. Lecture 22 and 23 are where this stuff ends up being really useful (not in MATH7021 unfortunately).

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Test 1

The first test is on this Thursday 25 October at 8.20 p.m.

Please find a sample test.

Note that the format will be the same as this.

1. Forward Difference Methods (14 marks)
2. Gaussian Elimination Methods including Partial Pivoting 7 marks)
3. The Jacobi and Gauss-Siedel Method (7 marks)
4. Laplace Transforms of Differential Equations (7 marks)

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Test 1

Consider this notice for the test on Thursday 25 October at 8.20 p.m (just under two weeks away) (I have spoken with Ted and I think there is going to be making a change to his assignment so for now we are full speed ahead for Week 6.  Note that there is still a small chance that this will be held in Week 7 Thursday 1 November at 8.20 p.m.).

Please find a sample test. I will give ye a copy of this Thursday night which will include the Finite Differences table and the Laplace Transform tables.

Note that the format will be the same as this.

1. Forward Difference Methods (14 marks)
2. Gaussian Elimination Methods including Partial Pivoting 7 marks)
3. The Jacobi and Gauss-Siedel Method (7 marks)
4. Laplace Transforms (7 marks)

Q. 4 will be covered Thursday night.

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Applications of Linear Algebra to Civil Engineering

A quick Google of ‘applications of linear algebra to civil engineering’; shows up a lot of the following.

• Truss systems give rise to linear systems — as does any analysis of equilibria
• Traffic Flow Problems
• Electrical Circuits
• Numerical Solution of Differential Equations (e.g. Beam Equations)
• Stress and Strain
• Elastic Systems — such as systems of springs and pulleys
• Torque

Also a lot of the theory of differential equations is best analysed using (infinite dimensional) linear algebra. For example, we said that the solutions of second order linear homogenous differential equations must be two dimensional. This is a linear algebra result.

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Exercise Answers

Section 1.1

Q. 8 , . Q. 9 , . Q. 10 , . Q. 11 , . Q. 12 , . Q. 14 . Q. 15 . Q. 16 , . Q. 17 and .

Section 1.3

To check that we have the correct solutions to simultaneous equations/linear systems we can plug in our values in to ALL of the equations and see that our solution set satisfies all of the equations. Note that solving two out of three equations does not mean that we have a solution. Quite often an arithmetic slip will give us a solution set that only fails for one of the equations. Hence I give solutions to the systems in terms of the coordinates .

Q. 2 (iv) , (v) for , (vi) , (vii) , (viii) , (ix) . Q. 4 All false.