This preview shows page 1. Sign up to view the full content.
264
CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
or
∂f
∂y
φ
′
(
x
) =
−
∂f
∂x
φ
′
(
x
) =
−
∂f/∂x
∂f/∂y
as required.
A mnemonic device to remember which partial goes on top of this
fraction and which goes on the bottom is to write Equation (
3.18
)
formally as
dy
dx
=
−
dy
dz
dz
dx
–that is, we formally (and unjustiFably) “cancel” the
dz
terms of the
two “fractions”. (Of course, we have to remember separately that we
need the minus sign up front.)
•
Equation (
3.18
) can also be interpreted as saying that a vector
tangent to the level curve has slope
φ
′
(
x
) =
−
b
∂f
∂x
(
x,φ
(
x
))
Bsb
∂f
∂y
(
x,φ
(
x
))
B
,
which means that it is perpendicular to
−→
∇
f
(
x,φ
(
x
)). Of course, this
could also be established using the Chain Rule (Exercise
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus

Click to edit the document details