The notion of a derivative has numerous interpretations and applications. A well-known geometric interpretation is that of a slope, or more generally that of a linear approximation to a mapping between linear spaces (see http://planetmath.org/node/2975here). Another useful interpretation comes from physics and is based on the idea of related rates. This second point of view is quite general, and sheds light on the definition of the derivative of a manifold mapping (the latter is described in the pushforward entry).
Consider two physical quantities and that are somehow coupled. For example:
the quantity could be the radius of a sphere and the sphere’s surface area;
the quantity could be the horizontal position of a point on a given curve and the distance traversed by that point as it moves from some fixed starting position;
the quantity could be depth of water in a conical tank and the rate at which the water flows out the bottom.
Regardless of the application, the situation is such that a change in the value of one quantity is accompanied by a change in the value of the other quantity. So let’s imagine that we take control of one of the quantities, say , and change it in any way we like. As we do so, quantity follows suit and changes along with . Now the analytical relation between the values of and could be quite complicated and non-linear, but the relation between the instantaneous rates of change of and is linear.
It does not matter how we vary the two quantities, the ratio of the rates of change depends only on the values of and . This ratio is, of course, the derivative of the function that maps the values of to the values of . Letting denote the rates of change of the two quantities, we describe this conception of the derivative as
or equivalently as
Next, let us generalize the discussion and suppose that the two quantities and represent physical states with multiple degrees of freedom. For example, could be a point on the earth’s surface, and the position of a point 1 kilometer to the north of . Again, the dependence of and is, in general, non-linear, but the rate of change of does have a linear dependence on the rate of change of . We would like to say that the derivative is precisely this linear relation, but we must first contend with the following complication. The rates of change are no longer scalars, but rather velocity vectors, and therefore the derivative must be regarded as a linear transformation that changes one vector into another.
In order to formalize this generalized notion of the derivative we must consider and to be points on manifolds and , and the relation between them a manifold mapping . A varying is formally described by a parameterized curve
The “coupling” of the two quantities is described by the composition
The derivative of at any given is a linear mapping
All of the above has a perfectly rigorous presentation in terms of manifold theory. The approach of the present entry is more informal; our ambition was merely to motivate the notion of a derivative by describing it as a linear transformation between velocity vectors.
|Date of creation||2013-03-22 12:44:59|
|Last modified on||2013-03-22 12:44:59|
|Last modified by||rmilson (146)|