## Thursday, October 13, 2011

### 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.