This preview shows page 1. Sign up to view the full content.
336
CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
We are looking for the extreme values of this function on the curve of
intersection of two level surfaces. In principle, we could parametrize the
ellipse, but instead we will work directly with the constraints and their
gradients:
g
1
(
x,y,z
) =
x
2
+
y
2
−→
∇
g
1
= (2
x,
2
y,
0)
g
2
(
x,y,z
) =
x
+
y
+
z
−→
∇
g
2
= (1
,
1
,
1)
.
Since our curve lies in the intersection of the two level surfaces
L
(
g
1
,
4) and
L
(
g
2
,
1), its velocity vector must be perpendicular to both gradients:
−→
v
·
−→
∇
g
1
= 0
−→
v
·
−→
∇
g
2
= 0
.
At a place where the restriction of
f
to this curve achieves a local (relative)
extremum, the velocity must also be perpendicular to the gradient of
f
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