Second Order Partial Derivatives

1
Second Order Partial Derivatives

• Curvature in Surfaces

2
The un-mixed partials fxx and fyy

• We know that fx(P) measures the slope of the
  graph of f at the point P in the positive x
  direction.
• So fxx(P) measures the rate at which this slope
  changes when y is held constant. That is, it
  measures the concavity of the graph along the
  x-cross section through P.

Likewise, fyy(P) measures the concavity of the
graph along the y-cross section through P.
3
The un-mixed partials fxx and fyy

• fxx(P) is
• Positive
• Negative
• Zero

Example 1
What is the concavity of the cross section along
the black dotted line?
4
The un-mixed partials fxx and fyy

• fyy(P) is
• Positive
• Negative
• Zero

Example 1
What is the concavity of the cross section along
the black dotted line?
5
The un-mixed partials fxx and fyy

• fxx(Q) is
• Positive
• Negative
• Zero

Example 2
What is the concavity of the cross section along
the black dotted line?
6
The un-mixed partials fxx and fyy

• fyy(Q) is
• Positive
• Negative
• Zero

Example 2
What is the concavity of the cross section along
the black dotted line?
7
The un-mixed partials fxx and fyy

• fxx(R) is
• Positive
• Negative
• Zero

Example 3
What is the concavity of the cross section along
the black dotted line?
8
The un-mixed partials fxx and fyy

• fxx(R) is
• Positive
• Negative
• Zero

Example 3
What is the concavity of the cross section along
the black dotted line?
9
The un-mixed partials fxx and fyy
Example 3

• Note The surface is concave up in the
  x-direction and concave down in the y-direction
  thus it makes no sense to talk about the
  concavity of the surface at R. A discussion of
  concavity for the surface requires that we
  specify a direction.

10
The mixed partials fxy and fyx

• fxy(P) is
• Positive
• Negative
• Zero

Example 1
What happens to the slope in the x direction as
we increase the value of y right around P? Does
it increase, decrease, or stay the same?
11
The mixed partials fxy and fyx

• fyx(P) is
• Positive
• Negative
• Zero

Example 1
What happens to the slope in the y direction as
we increase the value of x right around P? Does
it increase, decrease, or stay the same?
12
The mixed partials fxy and fyx
Example 2

• fxy(Q) is
• Positive
• Negative
• Zero

What happens to the slope in the x direction as
we increase the value of y right around Q? Does
it increase, decrease, or stay the same?
13
The un-mixed partials fxy and fyx

• fyx(Q) is
• Positive
• Negative
• Zero

Example 2
What happens to the slope in the y direction as
we increase the value of x right around Q? Does
it increase, decrease, or stay the same?
14
The mixed partials fyx and fxy

• fxy(R) is
• Positive
• Negative
• Zero

Example 3
What happens to the slope in the x direction as
we increase the value of y right around R? Does
it increase, decrease, or stay the same?
15
The mixed partials fyx and fxy

• fyx(R) is
• Positive
• Negative
• Zero

Example 3
?
What happens to the slope in the y direction as
we increase the value of x right around R? Does
it increase, decrease, or stay the same?
16
The mixed partials fyx and fxy

• fyx(R) is
• Positive
• Negative
• Zero

Example 3
What happens to the slope in the y direction as
we increase the value of x right around R? Does
it increase, decrease, or stay the same?
17
Sometimes it is easier to tell. . .

• fyx(R) is
• Positive
• Negative
• Zero

Example 4
W
What happens to the slope in the y direction as
we increase the value of x right around W? Does
it increase, decrease, or stay the same?
18
To see this better. . .
What happens to the slope in the y direction as
we increase the value of x right around W? Does
it increase, decrease, or stay the same?
Example 4

• The cross slopes go from
• Positive to negative
• Negative to positive
• Stay the same

W
19
To see this better. . .

• fyx(R) is
• Positive
• Negative
• Zero

Example 4
W