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Derivatives Vectors

Consider a vector, $\vec{p}$ , as a point in space. If that vector is a function of a real continuous parameter for instance, $t$ , then $\vec{p}(t)$ represents the loci as a function of a parameter.

If $\vec{p}(t)$ is continuous, then it sweeps out a continuous curve as $t$ changes continuously. It is very natural to think of $t$ as time and $\vec{p}(t)$ as the trajectory of a particle--such a trajectory would be continuous if the particle does not disappear at one instant, $t$ , and reappear an instant later, $t + dt$ , some finite distance distance away from $\vec{p}(t)$ .

If $\vec{p}(t)$ is continuous, then the limit:

$$\frac{d \vec{p}(t)}{dt} = \lim_{\Delta t \rightarrow 0} \frac{\vec{p}(t + \Delta t) - \vec{p}(t)} {\Delta t}$$ (11-5)

Notice that the numerator inside the limit is a vector and the denominator is a scalar; so, the derivative is also a vector. Think about the equation geometrically--it should be apparent that the vector represented by the derivative is locally tangent to the curve that is traced out by the points $\vec{p}(t - dt)$ , $\vec{p}(t)$ $\vec{p}(t + dt)$ , etc.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

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Vector Derivatives

Review: Partial and total derivatives

One might also consider a time- and space-dependent vector field, for instance $\vec{E}(x,y,z,t) = \vec{E}(\vec{x},t)$ could be the force on a unit charge located at $\vec{x}$ at time $t$ .

Here, there are many different things which might be varied and give rise to a derivative. Such questions might be:

1. How does the force on a unit charge differ for two nearby unit-charge particles, say at $(x,y,z)$ and at $(x,y + \Delta y, z)$ ?
2. How does the force on a unit charge located at $(x,y,z)$ vary with time?
3. How does the the force on a particle change as the particle traverses some path $(x(t),y(t),z(t))$ in space?

Each question has the ``flavor'' of a derivative, but each is asking a different question. So a different kind of derivative should exist for each type of question.

The first two questions are of the nature, ``How does a quantity change if only one of its variables changes and the others are held fixed?'' The kind of derivative that applies is the partial derivative.

The last question is of the nature, ``How does a quantity change when all of its variables depend on a single variable?'' The kind of derivative that applies is the total derivative. The answers are:

1. $$\ensuremath{\frac{\partial{\vec{E}(x,y,z,t)}}{\partial{y}}} = \en... ... \left( \frac{\partial{\vec{E}}}{\partial{y}} \right)_{\text{constant} x,z,t} }$$ (11-6)

   
   

2. $$\ensuremath{\frac{\partial{\vec{E}(x,y,z,t)}}{\partial{t}}} = \en... ... \left( \frac{\partial{\vec{E}}}{\partial{t}} \right)_{\text{constant} x,y,z} }$$ (11-7)

   
   

3. $$\frac{d\vec{E}(x(t),y(t),z(t),t)}{dt} = \ensuremath{\frac{\partia... ...\cdot \frac{d \vec{x}}{dt} + \ensuremath{\frac{\partial{\vec{E}}}{\partial{t}}}$$ (11-8)

   
   

All vectors are not spatial

It is useful to think of vectors as spatial objects when learning about them--but one shouldn't get stuck with the idea that all vectors are points in two- or three-dimensional space. The spatial vectors serve as a good analogy to generalize an idea.

For example, consider the following chemical reaction:

Reaction:	H$_{2}$	$\ensuremath{\frac{1}{2}}$O$_{2}$	$\rightleftharpoons$	$\ensuremath{\mbox{H}_2\mbox{O}}$
Initial:	1	1	$\rightleftharpoons$	0
During Rx.:	$1 - \xi$	$1 - \ensuremath{\frac{1}{2}}\xi$	$\rightleftharpoons$	$\xi$

The composition could be written as a vector:

$$\vec{N}= \left( \begin{array}{l} \text{moles } \mbox{H}_{2} \ \t... ...{array}{l} 1 - \xi\ 1 - \ensuremath{\frac{1}{2}}\xi \ \xi \end{array} \right)$$ (11-9)

and the variable $\xi$ plays the role of the ``extent'' of the reaction--so the composition variable $\vec{N}$ lives in a reaction-extent ($\xi$ ) space of chemical species.