Mathematical Tools for Physics

8—Multivariable Calculus 212

this Leibnitz notation is designed to lead you in the right direction, so it’s not an accident that itlookslike you’re
just canceling thedt’s.
Apply this to the differential offfromEq. ( 5 ).

df=df

(

x(t),y(t),dx(t,dt),dy(t,dt)

)

=

(

∂f

∂x

)

y

dx+

(

∂f

∂y

)

x

dy

=

(

∂f

∂x

)

y

dx

dt

dt+

(

∂f

∂y

)

x

dy

dt

dt

The derivativedf/dtis the coefficient ofdtin this differential, just as in Eq. ( 3 ).

df

dt

=

(

∂f

∂x

)

y

dx

dt

+

(

∂f

∂y

)

x

dy

dt

(7)

Example: (When you want to check out an equation, you should construct an example so that it reveals a
lot of structure without requiring a lot of calculation.)

f(x,y) =Axy^2 , and x(t) =Ct^3 , y(t) =Dt^2

First do it using the chain rule.

df

dt

=

(

∂f

∂x

)

y

dx

dt

+

(

∂f

∂y

)

x

dy

dt

=

(

Ay^2

)(

3 Ct^2

)

+

(

2 Axy

)(

2 Dt

)

=

(

A(Dt^2 )^2

)(

3 Ct^2

)

+

(

2 A(Ct^3 )(Dt^2 )

)(

2 Dt

)

= 7ACD^2 t^6

Now repeat the calculation by first substituting the values ofxandyand then differentiating.

df

dt

=

d

dt

[

A(Ct^3 )(Dt^2 )^2

]

=

d

dt

[

ACD^2 t^7

]

= 7ACD^2 t^6