36600 4
MA LECTURE NOTES: MONDAY, FEBRUARY 2
Total Dierentials. Say that z = f (x, y ) is a function. We will show that
f
f
dz =
dx +
dy.
x
y
This is called the total dierential of the function z = f (x, y ). First we make sense of this notation. Say
that we have a path = (t) in the plane R2 . That is, given any t R, we have a point (x, y ) = (t) R2 .
Then the composition
z (t) = f (t)
is actually a function of t. Choose a small dierence t, and dene the dierences x and y by
(x, y ) = (t + t) (t)
i.e.,
(x + x, y + y ) = (t + t).
Figure 4. Plot of (t) vs. t
3.2
2.4
1.6
0.8
-4.8
-4
-3.2
-2.4
-1.6
-0.8
0
0.8
1.6
2.4
3.2
4
4.8
-0.8
-1.6
-2.4
-3.2
We have the ratio
f + (t t) f (t)
z (t + t) z (t)
=
t
t
f (x + x, y + y f (x, y )
=
t
f (x + x, y + y ) f (x, y + y ) f (x, y + y ) f (x, y )
=
+
t
t
f (x + x, y + y ) f (x, y + y ) x f (x, y + y ) f (x, y ) y
=
+
.
x
t
y
t
Upon taking limits, we nd that
dz
f (x + x, y ) f (x, y ) x f (x, y + y ) f (x, y ) y
f dx f dy
= lim
+
=
+
.
t0
dt
x
t
y
t
x dt
y dt
This is called the total derivative of the function z = f (x, y ). Note that it involves partial derivatives. This
formula is the same as the more familiar result involving the dot product of the gradient:
dz
f f
= (f ) (t)
in terms of
f (x, y ) =
,
.
dt
x y
Upon multiplying both sides by the dierential dt we nd the total dierential above.