Sometimes several variables are related and, given information about how one variable is changing, you
want to find how the others are changing.
Example 12.7.1Bernoulli’s law states that in an incompressible fluid,
v2 P
2g + z +-γ = C
where C is a constant. Here v is the speed, P is the pressure, and z is the height above some referencepoint. The constants g and γ are the acceleration of gravity and the weight density of the fluid. Supposemeasurements indicate that
dv
dt
= −3, and
dz
dt
= 2. Find
dP
dt
when v = 7 and z = 8 in terms of g andγ.
This is just an exercise in using the chain rule. Differentiate the two sides with respect to
t.
1 dv dz 1 dP
gv dt + dt + γ-dt-= 0.
Then when v = 7 and z = 8, finding
dP-
dt
involves nothing more than solving the following for
dP
dt
.
7 1-dP--
g (− 3) +2 + γ dt = 0
Thus
( )
dP- = γ 21− 2
dt g
at this instant in time.
Example 12.7.2In Bernoulli’s law above, each of v,z, and P are functions of
(x,y,z)
, the positionof a point in the fluid. Find a formula for
∂P∂x-
in terms of the partial derivatives of the other variables.
This is an example of the chain rule. Differentiate both sides with respect to x.
v 1-
gvx + zx + γPx = 0
and so
( )
P = − vvx +-zxg γ
x g
Example 12.7.3Suppose a level curve is of the form f
(x,y)
= C and that near a point on thislevel curve y is a differentiable function of x. Find
dy
dx
.
This is an example of the chain rule. Differentiate both sides with respect to x. This gives
f + f dy-= 0.
x ydx
Solving for
ddyx
gives
dy − f (x,y)
---= ---x-----.
dx fy(x,y)
Example 12.7.4Suppose a level surface is of the form f
(x,y,z)
= C. and that near a point
(x,y,z)
on this level surface z is a C^{1}function of x and y. Find a formula for z_{x}.
This is an example of the use of the chain rule. Differentiate both sides of the equation with respect to
x. Since y_{x} = 0,
fx + fzzx = 0.
Then solving for z_{x},
z = −-fx(x,y,z)
x fz(x,y,z)
Example 12.7.5Polar coordinates are
x = rcosθ,y = rsin θ. (12.13)
(12.13)
Thus if f is a C^{1}scalar valued function you could ask to express f_{x}in terms of the variables r and θ. Doso.
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 f
(x,y)
= f
(r,θ)
and
so
fx = frrx + fθθx.
This will be done if you can find r_{x} and θ_{x}. 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
(12.13),