Math - Fluid Dynamics - Vectors - Functions related to vectors and scalars

In order to locate a position in space and time, one may use the position vector as a function of time which happens to be a scalar value.  This relation is an example of a vector as a function of a scalar.

\[ \mathbf{r} = \mathbf{r} \left( t \right) \]

or in general if for each scalar variable there exist a vector value then

\[ \mathbf{A} = \mathbf{A} \left( t \right) \]

This relation can be inversed so that a scalar is a function of a vector.  One example is temperature which can be described at every point so that

\[ T = T \left( \mathbf{r} \right) \]

or, again, in general if for each vector value there exist a scalar value

\[ \phi = \phi \left( \mathbf{r} \right) \]

In the case when \( \mathbf{r} \) is the position vector, then the scalar is a function of position.

A third example denotes a vector as a function of a vector.  If one looks at the rigid body rotation that rotates at a constant angular velocity, \( \boldsymbol{\omega} \), then the velocity at a point on the body can be described as

\[ \mathbf{V} = \boldsymbol{\omega} \times \mathbf{r} \]

where the position vector, \( \mathbf{r} \), is taken from the axis of rotation.

Thus, the velocity vector, \( \mathbf{V} \), becomes a function of the position vector, \( \mathbf{r} \).

\[ \mathbf{V} =  \mathbf{V} \left( \mathbf{r} \right) \]

Likewise, the general form becomes

\[ \mathbf{A} =  \mathbf{A} \left( \mathbf{r} \right) \]

and when \(  \mathbf{r} \) is the position vector \(  \mathbf{A} \) is a vector function of position.