5. THE CHAIN RULE
93
Example. Suppose z = f (x, y) with x = r cos ✓, y = r sin ✓. Find fr , frr .
We have
@x
@y
= cos ✓ and
= sin ✓.
@r
@r
fr = fx ·
@x
@y
+ fy ·
= fx cos ✓ + fy sin ✓
@r
@r
@
@
(fr ) =
fx cos ✓ + fy sin ✓
@r
@r
@
@
= (fx) cos ✓ + (fy ) sin ✓
@r
@r
⇣
⌘
⇣
@x
@y
=
fxx
+ fxy
cos ✓ +
@r
@r
|
{z
}
frr =
Use fx for f in diagram above
@x
@y
fyx
+ fyy
| @r {z @r}
Use fy for f in diagram above
⇣
⌘
⇣
⌘
= fxx cos ✓ + fxy sin ✓ cos ✓ + fyx cos ✓ + fyy sin ✓ sin ✓
= fxx cos2 ✓ + 2fxy sin ✓ cos ✓ + fyy sin2 ✓
⌘
sin ✓