Vector field and scalar field

There is a very common confusion between a scalar field and vector field, mainly because of the definitions of these.

Thus we call the vector field, mathematically speaking, a function whose domain and range are subsets of three-dimensional Euclidean space $ latex R3$.

Let’s begin by shedding the definition we have just given. A vector field F, represented by \overrightarrow F, associates a vector \overrightarrow F (x, y, z) at each point (x, y, z). Going depper a bit to something more known, we say that at each point of space (x, y, z) we associate a vector that will be function of that point and therefore, we can define it as $ latex \overrightarrow {F}(x , Y, z)$. What is now easier to understand.

We have that each of the components of \overrightarrow F are scalar functions F_{1} (x, y, z), F_{2} (x, y, z), F_{3} (x , Y, z) y, so \overrightarrow F (x, y, z) can be expressed as a function of the canonical basis of R_{3} thus:

\overrightarrow F(x,y,z)= F_{1}(x, y, z)\overrightarrow i + F_{2}(x, y, z)\overrightarrow j + F_{3}(x, y, z)\overrightarrow k

And what is the scalar field? Obviously, each of the components of the vector field is a scalar field. That is, the amount and how we obtain or vary in each of the components is nothing more than a scalar field. That is much simpler to understand.

Example of a vector field is the gravitational field, for example and its values ​​is an example of scalar field.

By way of summary and trying to facilitate to the maximum thus its understanding we could say that a vector field represents in each point of the space a vector whereas a scalar field represents in each point of the space a value. An example of a vector field would be the gravity or velocities of a body rotating through an axis, (eg the Z, as the image accompanying this post) however, a scalar field would represent temperatures in a room or space .

From this last example of scalar field, how it would vary or be distributed over time could be a vector field since we would have vectors of “movement” of the heat or, in a closed space, the values ​​of the gravitational attraction ) Would be a scalar field.

Do you indicate that the vector and scalar fields are interchangeable? No. What I indicate is that they are related since, if we look at the definition, a vector field, each of its components are values ​​of a scalar field.