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.