Vectorial and scalar fields

In this post I want to perform several commentaries about this basic concepts that it is not trivial for a beginner in the world of mathematics. I'm refering to scalar and vectorial fields.

It's usual that once you have finished your graduate on engineering, your concept about scalar and vectorial fields is basically the direction and sign of the magnitude. This is true, but when we are facing a rigurous work with equations it is not very useful. I will explain it more in detail, and will include several examples related with fluid mechanics, a classical field of study where both vectorial and scalar functions are used.

Scalar field: Scalar fields are those which can be defined as combination of independent variablesrelated with the system of reference. When these functions are evaluated in a point of the domain in which they are defined, they give as a result a number (2, 0.1, 7E-6). Typical examples could be the temperature or the pressure: they have a distribution in the space following a scalar field dependent only of the space and time.

Example: Pressure in a point

$$ P(x,y,z)=4x^2+3y^2-8.4xy-2z+1 $$

As it can be seen, pressure is a function of space coordinates x,y and z. Its valus only depend of the position we choose.

Vectorial field: These type of fields need for they definition a vectorial basis, that is, they have to be refered to a system of coordinates. These magnitudes habe different values for a single point depending which direction we consider (these are called "vector components"). Each vector component will be a function of other variables of the system. Notice that, as we are using a basis, direction and sign appears naturally associated with vectorial fields.


Example: Velocity of a fluid particle in a region.

$$u=[u_x,u_y,u_z]=\left[4x^2-9y+z,3z^4-2,2x+4y^3\right]$$

As it can be seen, u vector have three components, according with space dimension in which field is described. Each component is at the same time a function of x, y and z. Global velocity of each particle will be the sum of all components, but in a vectorial way: modulus will be (in cartesian coordinates)  the square root of the squared sum of each component, and direction will be the respective unitary vector representing direction and sign.

Maybe this difference is obvious, but I think it worth to mention here: this will suppose a difference when we start to practice tensorial calculus.

In Further posts I will write a small table with different results of operators calculated for tensorial and scalar magnitudes.

Latex writting for scientific texts

Hello everybody


For those who make use of Latex in your scientific documents but don't know how to upload them properly on your blog or website, I want to inform about a webpage in which it is explained easily how to do it:

http://watchmath.com/vlog/?p=438

The process is very easy. Just copy the script that appear in the author's web, but changing only a line. You should remove

http://www.watchmath.com/cgi-bin/mathtex3.js

and copy instead:

http://www.watchmath.com/main/cgi-bin/mathtex3.js

That's it ! Latex on your blog.

If you are not familiar with Latex format, I will give you some examples of what kind of format generate. In case you are interested on it, just google it. There are thousand of webs and manuals of how to use Latex. It requires some efforts to learn the basic commands, but once it is done the system is much powerful that conventional word processors when writting scientific notation.  Actually I'm not using word anymore!!

$$ F(x,y)=\int_{a}^{b} g(x,y) dx $$
$$\nabla \cdot \gamma = \frac{\partial \gamma}{\partial x}+ \frac{\partial \gamma}{\partial y} $$


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Basic details of Divergence and Curl (I)

For those whe study tensor mathematics these concepts are already familiar. However, when should have to realize that they also have a physical meaning when applied to particular cases.

- Divergence is always related with normal component of a certain flow trespassing the surface of a certain volume (variation of a certain magnitude per unit area and time). A direct consequence of this imply that if we measure the income and outcome contributions of this magnitude through its surface, this balance will be directly related with the value of divergence: If divergence is zero the net contribution of magnitude in this volume is exactly zero (the same amount is incoming that outcoming). If divergence is positive more flux of magnitude is outcoming in the selected region that incoming, and negative divergence will imply the opposite.

A classic example would be the magnetic field, a vectorial field. If we take a control volume around a magnet it can be seen that divergence of the magnetic field is zero. Knowing that a magnet is in fact a source of the magnetic field, zero divergence tell us that "streamlines" of the field must be closed: streamlines are generated in a point of the magnet, but they must flow to the other pole of the magnet. 

Other well known example is fluid mechanics. Zero divergence around a control volume imply mass conservation, and flow conservation with almost incompressible fluids like water.

It is remarkable that the divergence of a vectorial field is a SCALAR.

Curl also have a specific meaning when we are talking about vectorial magnitudes. This operator is linked with tangential component. To say some examples, in fluid mechanics curl operator appear when we define the concepts "flow circulation " and "vorticity". The result of applying curl to two vectorial field is another vectorial field orthogonal to the previous ones.

About me

Hello world

In this post I would like to introduce myself and talk a little bit more about me and what is my objective with this blog.

Mi name is David Rodriguez. I got my degree on Chemical Engineering in Universidad de Zaragoza and I also have a master degree on Process Engineering in Universidad Complutense de Madrid. Nowadays I work as a researcher in an aeronautics project. I share my work with additional courses included in a master of industrial mathematics in Universidad Carlos III de Madrid and with my PhD on applied mathematics that, if everything is fine, I will get on Autumn 2012.

I consider myself a very dynamic and adaptable person, very interested on science and technology. I'm very patient with my work and concerned about doing my best all the time. I hate leave my tasks unfinished.

I'm always ready to learn new things, but I don't leave nobody decide instead of me. I have my own criteria and fashion is not for me. My character is quite defined since I was a child, and I'm aware of that: I act based on it with all consequences.

I think my parents gave me an excellent education, because now I'm independent and able to take my decisions without being influenced much by modern marketing and social pressure.

I practice martial arts and I really love mountain, specially trekking. And I'm also a lover of video games, as part of the generation which saw the raising of this powerful industry.

I like theatre, cinema, going out with my friends, listen music and practice whatever sport I have the chance. I try to see documentaries about science and society: I think it's important to know in what kind of world we are living in, and many times new ideas comes from unspected places...

This blog has been created with the purpose of sharing my opinions and knowledge with anybody interested on the topics I will talk here. Basically,  my purpose is to focus my efforts on writing about very nice mathematics I've learned during this past two years (and I'm still learning).

I hope my posts will be useful for those who try to read them

Greetings


DAVID