Meaning of derivatives of vector fields?

Question

I have a doubt about the real meaning of the derivative of a vector field. This question seems silly at first but the doubt came when I was studying the definition of tangent space. If I understood well a vector is a directional derivative operator, i.e.: a vector is an operator that can produce derivatives of scalar fields. If that's the case then a vector acts on a scalar field and tells me how the field changes on that point. However, if a vector is a derivative operator, a vector field defines a different derivative operator at each point. So differentiate a vector would be differentiate a derivate operator, and that seems strange to me at first. I thought for example that the total derivative of a vector field would produce rates of change of the field, but my studies led me to a different approach, where the total derivative produces rates of change only for scalar fields and for vector fields it produces the pushforward. So, what's the real meaning of differentiating a vector field knowing all of this?

Answer 1

Greetings! I know what you mean because studying vectors fields is part of my research in Calculus. It is very interesting indeed! Taking the derivative of vector fields creates more vector fields going in different directions. Then, eventually, the vector fields all become scalar vector fields.

Continuous derivatives of vectors fields creates something called a manifold and is coined "smooth' due to the continuous differentiation.

Let me know if this helped some.

Answer 2

It doesn't seem you are describing vectors in a very common way. One can have differential operators, and apply them to vector fields, but (to my knowledge) a vector is not "an operator that can produce derivatives of scalar fields".