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## Summary:

- The Derivative of a scalar field at a point is normally described as the "cotangent vector of the field at that point". Can we define the derivative of the field at a point, for a constant scalar field, as the "linear density" of the field at the point?

Wikipedia defines the derivative of a scalar field, at a point, as the cotangent vector of the field at that point.

In particular;

The gradient is closely related to the derivative, but it is not itself a derivative: the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a

Now if a scalar fields value varies with x according to F = f(x) then the derivative is clearly f` (x) which may be different from zero (depending in the form of f(x)).

If the field has a constant value, over x, e.g Field = 4x. Can we say that the derivative = 4 and represents the "linear density" of the Field along the x axis ? In such a case the Gradient will be a vector (having magnitude = 4) being coincident with the x-axis. The Directional derivative of the field, at that point, will only exist along the x-axis (except as we may define values for the Field along the Y and Z axis) and will also have, in this case, a constant value of 4 in the direction of the x-axis.

I am only accustomed to thinking of the above in the case where f(x) is a function whose derivative is not a constant value.

Hope this question makes sense ? If I am correct then the answer to the above is trivial but will clarify things in my own mind.

In particular;

The gradient is closely related to the derivative, but it is not itself a derivative: the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a

*co*tangent vector .Now if a scalar fields value varies with x according to F = f(x) then the derivative is clearly f` (x) which may be different from zero (depending in the form of f(x)).

If the field has a constant value, over x, e.g Field = 4x. Can we say that the derivative = 4 and represents the "linear density" of the Field along the x axis ? In such a case the Gradient will be a vector (having magnitude = 4) being coincident with the x-axis. The Directional derivative of the field, at that point, will only exist along the x-axis (except as we may define values for the Field along the Y and Z axis) and will also have, in this case, a constant value of 4 in the direction of the x-axis.

I am only accustomed to thinking of the above in the case where f(x) is a function whose derivative is not a constant value.

Hope this question makes sense ? If I am correct then the answer to the above is trivial but will clarify things in my own mind.