The vector field you are talking about is actually a generalized quantity field.

Such as a three-dimensional vector field in a three-dimensional space.

It can also be directly divided into three three-dimensional quantitative fields.

The gradient can be obtained, and of course the gradient field is also a vector field.

Physical meaning is an abstract concept of human beings, depending on the specific situation.

Just like fractions, roots and complex numbers are all inventions of human beings to solve specific problems.

For example, I take the partial derivative of the three-dimensional vector field of (u, v, w) velocity about (x, y, z).

You get a tensor field.

If you have studied theoretical mechanics or material mechanics or fluid mechanics, even if you are only a math student.

You should all know the tensor.

The tensor field of velocity field can be decomposed into the sum of strain rate tensor and rotation tensor.

Different physical explanations in different situations.

Inventing what you need, nothing is impossible. I believe that people who study engineering know better what is practical.