related rates

The notion of a derivative has numerous interpretations andapplications. A well-known geometric interpretation is that of aslope, or more generally that of a linear approximation to a mappingbetween linear spaces (see http://planetmath.org/node/2975here). Another usefulinterpretation comes from physics and is based on the idea of relatedrates. This second point of view is quite general, and sheds light onthe definition of the derivative of a manifold mapping (the latter isdescribed in the pushforward entry).

Consider two physical quantities $x$ and $y$ that are somehow coupled.For example:

• the quantities $x$ and $y$ could be the coordinates of a point as itmoves along the unit circle;

• the quantity $x$ could be the radius of a sphereand $y$ the sphere’s surface area;

• the quantity $x$ could be the horizontal position of a point ona given curve and $y$ the distance traversed by that point as itmoves from some fixed starting position;

• the quantity $x$ could be depth of water in a conical tank and$y$ the rate at which the water flows out the bottom.

Regardless of the application, the situation is such that a change inthe value of one quantity is accompanied by a change in the value ofthe other quantity. So let’s imagine that we take control of one ofthe quantities, say $x$, and change it in any way we like. As we doso, quantity $y$ follows suit and changes along with $x$. Now theanalytical relation between the values of $x$ and $y$ could be quitecomplicated and non-linear, but the relation between the instantaneousrates of change of $x$ and $y$ is linear.

It does not matter how we vary the two quantities, the ratio of therates of change depends only on the values of $x$ and $y$. This ratiois, of course, the derivative of the function that maps the values of$x$ to the values of $y$. Letting $\dot{x},\dot{y}$ denote the ratesof change of the two quantities, we describe this conception of thederivative as

$$\frac{dy}{dx}=\frac{\dot{y}}{\dot{x}},$$

or equivalently as

$$\dot{y}=\frac{dy}{dx}\dot{x}.$$(1)

Next, let us generalize the discussion and suppose that the twoquantities $𝐱$ and $𝐲$ represent physical states with multipledegrees of freedom. For example, $𝐱$ could be a point on theearth’s surface, and $𝐲$ the position of a point 1 kilometer to thenorth of $𝐱$. Again, the dependence of $𝐲$ and $𝐱$ is, ingeneral, non-linear, but the rate of change of $𝐲$ does have alinear dependence on the rate of change of $𝐱$. We would like tosay that the derivative is precisely this linear relation, but we mustfirst contend with the following complication. The rates of change areno longer scalars, but rather velocity vectors, and therefore thederivative must be regarded as a linear transformation that changesone vector into another.

In order to formalize this generalized notion of the derivative wemust consider $𝐱$ and $𝐲$ to be points on manifolds $X$ and $Y$,and the relation between them a manifold mapping $\varphi :X\to Y$. A varying $𝐱$ is formally described by a parameterized curve

$$\gamma :I\to X,I\subset ℝ.$$

The correspondingvelocities take their value in the tangent spaces of $X$:

$${\gamma }^{\prime }(t)\in {\mathrm{T}}_{\gamma (t)}X.$$

The “coupling” of the two quantities is described by the composition

$$\varphi \circ \gamma :I\to Y.$$

The derivative of $\varphi$ at any given $𝐱\in X$ is a linear mapping

$${\varphi }_{*}(𝐱):{\mathrm{T}}_{𝐱}X\to {\mathrm{T}}_{\varphi (𝐱)}Y,$$

called thepushforward of $\varphi$ at $𝐱$, with the property that for everytrajectory $\gamma$ passing through $𝐱$ at time $t$, we have

$${(\varphi \circ \gamma )}^{\prime }(t)={\varphi }_{*}(𝐱){\gamma }^{\prime }(t).$$

The above is the multi-dimensional and coordinate-free generalizationof the related rates relation (1).

All of the above has a perfectly rigorous presentation in terms ofmanifold theory. The approach of the present entry is more informal;our ambition was merely to motivate the notion of a derivative bydescribing it as a linear transformation between velocity vectors.