21.7.1 Related Rates Problems
Sometimes several variables are related and, given information about how one variable is
changing, you want to find how the others are changing.
Example 21.7.1 Bernoulli’s law states that in an incompressible fluid,
where C is a constant. Here v is the speed, P is the pressure, and z is the height above
some reference point. The constants g and γ are the acceleration of gravity and the weight
density of the fluid. Suppose measurements indicate that
= 7 and z
= 8 in terms of g and γ.
This is just an exercise in using the chain rule. Differentiate the two sides with respect
Then when v = 7 and z = 8, finding
involves nothing more than solving the following
at this instant in time.
Example 21.7.2 In Bernoulli’s law above, each of v,z, and P are functions of
, the position of a point in the fluid. Find a formula for
in terms of the
partial derivatives of the other variables.
This is an example of the chain rule. Differentiate both sides with respect to
Example 21.7.3 Suppose a level curve is of the form f
C and that near
a point on this level curve y is a differentiable function of x. Find
This is an example of the chain rule. Differentiate both sides with respect to x. This
Example 21.7.4 Suppose a level surface is of the form f
C. and that
near a point
on this level surface z is a C1 function of x and y. Find a
formula for zx.
This is an example of the use of the chain rule. Differentiate both sides of the
equation with respect to x. Since yx = 0,
Then solving for zx,
Example 21.7.5 Polar coordinates are
Thus if f is a C1 scalar valued function you could ask to express fx in terms of the
variables r and θ. Do so.
This is an example of the chain rule. Abusing notation slightly, regard f as a function
of position in the plane. This position can be described with any set of coordinates. Thus
This will be done if you can find rx and θx. However you must find these in terms of r
and θ, not in terms of x and y. Using the chain rule on the two equations for the
transformation in 21.13,
Solving these using Cramer’s rule,
Hence fx in polar coordinates is