Second Order
Partial Derivatives
Curvature in Surfaces
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.
The “un-mixed” partials: fxx and fyy
Example 1
fxx(P) is
Positive
Negative
Zero
What is the concavity of the cross
section along the black dotted line?
The “un-mixed” partials: fxx and fyy
Example 1
fyy(P) is
Positive
Negative
Zero
What is the concavity of the cross
section along the black dotted line?
The “un-mixed” partials: fxx and fyy
Example 2
fxx(Q) is
Positive
Negative
Zero
What is the concavity of the cross
section along the black dotted line?
The “un-mixed” partials: fxx and fyy
Example 2
fyy(Q) is
Positive
Negative
Zero
What is the concavity of the cross
section along the black dotted line?
The “un-mixed” partials: fxx and fyy
Example 3
fxx(R) is
Positive
Negative
Zero
What is the concavity of the cross
section along the black dotted line?
The “un-mixed” partials: fxx and fyy
Example 3
fxx(R) is
Positive
Negative
Zero
What is the concavity of the cross
section along the black dotted line?
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.
The mixed partials: fxy and fyx
Example 1
fxy(P) is
Positive
Negative
Zero
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?
The mixed partials: fxy and fyx
Example 1
fyx(P) is
Positive
Negative
Zero
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?
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?
The “un-mixed” partials: fxy and fyx
Example 2
fyx(Q) is
Positive
Negative
Zero
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?
The mixed partials: fyx and fxy
Example 3
fxy(R) is
Positive
Negative
Zero
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?
The mixed partials: fyx and fxy
Example 3
fyx(R) is
?
Positive
Negative
Zero
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?
The mixed partials: fyx and fxy
Example 3
fyx(R) is
Positive
Negative
Zero
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?
Sometimes it is easier to tell. . .
fyx(R) is
Example 4
W
Positive
Negative
Zero
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?
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
W
The “cross” slopes go from
Positive to negative
Negative to positive
Stay the same
To see this better. . .

Example 4
W



fyx(R) is
Positive
Negative
Zero