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 , associates a vector 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 are scalar functions y, so can be expressed as a function of the canonical basis of thus:

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.

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