VIDEO: HOW TO SOLVE A DOUBLE INTEGRAL IN POLARS!

Here I bring you the video I told, I have had so much problems uploading but finally here it is!

MERRY CHRISTMAS!

Useful tips when integrating

As these days we´ve been reviewing integration, I think that is a good idea to bring to you some tips about integrating , that will help in order to not to make silly mistakes.

The first thing that is very recommendable is to sketch the region you are given. With a very simple sketch we will see much more clearly what the domain of the figure is and realize what we were asked for.

When we have this sketch, we get the bounds easily, so we write them as inequalities (<, >, ≤ and ≥), making sure that, at least, one of the variables has constant limits. And then we build up our integral, writing at the top and at the bottom the bounds previously calculated.

When you write this, note on the order you are integrating, because there is going to be almost always some variable with constant limits, and that variable therefore has to be the one on the outside integral.

It is recommendable also, to write the integration variable involved explicitly under each integral sign, this is to remind us which of x and y is considered a constant and which is considered a variable.

It is very helpful to learn by heart some easy formulas, to gain some precious time that will be very important in exams. (I will put here a link since I can´t write any math symbology here)

http://www.mathportal.org/calculus/integrals/integration-formulas.php

HOPE IT WAS HELPFUL AND SEE YOU IN THE NEXT POST THAT WILL BE A HOME-MADE VIDEO!

The Coastline Paradox!

Hi everyone! I´m posting one week more for bring to you freaky math facts and paradoxes. This time I bring a curious paradox that has something related with physics also, it is commonly called The Coastline Paradox.

Lewis Fry Richardson was trying to search for a relation between the probability of two countries going to war and the length of their common border, and then, he realized that there were big differences between the published lengths of international borders, so he started to think about what could be the problem. He concluded that, there were those differences due to The Coastline Paradox, also known as the Richardson effect. It postulates that the length of a landmass, mathematically speaking, it´s never well defined, because it depends on the length of the instrument that you are using to measure.

To explain the phenomena suppose a printed map of Spain, and then try to measure it with a 30 cm rule, it will result easy to measure, but it will give you a number that is far from the actual one. Now, try to measure it with the first centimetre of the rule, it´s a hard task, but the resulting number would be much more accurate than the other. The fact is that, the smaller the “rule” the more accurate the result, so with an infinite small number, the length would be infinite also!
This gif may help!

http://www.freeimagehosting.net/newuploads/41259.gif

INTERVIEW!

Hi it´s Gonzalo, and I am here with José María Martínez, a graduate in physics and mathematics, and I´m going to ask him several questions about the subject.

-Q: First of all, thanks for your time, I know that you don´t have much time so I appreciate this a lot. And I would like to start asking you if the education has change over the years since you teach.

A: Yes, of course, now the students are more prepared to possible situations at their professional lives, since the teaching is much more practical and less theoretical. There are many who disappoint in this, but I´m sure that this will be the best for your careers.

-Q: Do you agree with those people who are now manifesting for a greater public education?

A: In part yes, because we all need to make much more accessible education, in order to have an advanced population, the bad point is that with all the strikes, the losers are the students, because they lose many days of learning. What is certain is that without the strikes, the government would not respond.

-Q: What do you think about the exams in order to pass to university?

A: Well, I don’t think that´s the best way to test a student´s knowledge, because often is unfair for those examined, but they have the chance to get a good grade in bachillerato, so I do not consider too bad way at all.

-Q: What do you think of the recent government proposal to reduce the Erasmus scholarships?

A: I think that is the worst error they can make. The Erasmus Scholarships are extremely helpful for our students, who get the first international contact there and this will be very important in their futures, as it opens new workplaces. But, fortunately, mister Wert didn´t bring it forward.

And this is all, hope you liked it!

FREE CHOCOLATE!

Recently, I have seen a video in which it was told how to "get free chocolate", I was amazed about that paradox, so I started to investigate for myself, then I realized that it could be a good idea to post what I learnt about the subject, so here I bring you a bit of information about the "Chocolate Paradox".

In the video we can see that a man cuts a chocolate bar in two parts, but one different from the other, he starts to cut in the second  square in the left, and finishes in the third one from the other side. Then he cuts a vertical row from the left side, and finally, he cuts the last square of this row. Then he put the right piece on the left, and the cut row on the right, and there you have it! One "new" square of chocolate!

GIF that shows the cuts from a different way!

This of course has an explanation, because of course we are not creating chocolate from nothing! It seems that the chocolate bar, once we cut it, is the same, but with each cut we do, the bar decreases a little, just a few millimetres from each square of the last horizontal row. This image explains very well what is happening:

This peculiar phenomenon is part of the Geometric Paradoxes of Mathematics, where you can find another interesting paradoxes of shapes, surfaces and areas...

Hope you enjoy the video and the info given!

Euler number (e)

For my first post, I bring to you some information of one of the most famous special numbers on Mathematics, the number of Euler, most known as number e. This number appears many times, mostly of them to makes our lifes more difficult, it´s scary when you see it, because that means that the problem will turn a little bit more difficult, but once you´ve practice with it, it can be your friend when resolving a problem.

The number of Euler is a mathematician constant, it´s discovery was credited to Jacob Bernoulli, but the constant then was symbolized with the letter b. Was Leonhard Euler who started representing it with the letter e.

It has infinite decimals, as it is irrational, but the first decimals are easy to remember, here you have a little trick. The first three numbers are well known (2.71). Then you have the number 1828 that it appears doubled and finally you have the angles of a Right-angled isosceles triangle, which are 45º, 90º, 45º. Then you´ve got: e=2.7118281828459045…

An interesting property of the number e, if you divide a number into parts, and then you multiply those parts together, you´re going to get a number, but, did you know that the closer that number is to the e number (2.71), the bigger the final number is? Let´s demonstrate it with an example:

                -We have the number 12 for instance, then we divide it into equal parts: 20/4=5, then we do five to the power of four: 5x5x5x5=1024. But if we want the highest number that that formula could give, we have to divide our number (20) with a number that gives as result of the division a number close to the Euler number:

                                               20/8=2.5

                                               2.5^8=1525.87…

But if we, instead of 8, we divide 20 by for example 9, we´ve got 2.22…

                                               Then 2.22^9=1309.715

I hope you found these info at least interesting or useful!

WELCOME!

I welcome you to this blog, where you will find interesting info about Maths, or other subjects related with Maths, this blog is part of a project of the UE(Universidad Europea)´s calculus class. I will post several information and it will be as well as in video format or as links, interviews and images. Hope you´ll enjoy it as I do!