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 fyyfxx(P) is

Positive

Negative

Zero

Example 1

What is the concavity of the cross section along the black dotted line?

The “un-mixed” partials: fxx and fyyfyy(P) is

Positive

Negative

Zero

Example 1

What is the concavity of the cross section along the black dotted line?

The “un-mixed” partials: fxx and fyyfxx(Q) is

Positive

Negative

Zero

What is the concavity of the cross section along the black dotted line?

Example 2

The “un-mixed” partials: fxx and fyyfyy(Q) is

Positive

Negative

Zero

What is the concavity of the cross section along the black dotted line?

Example 2

The “un-mixed” partials: fxx and fyyfxx(R) is

Positive

Negative

Zero

Example 3

What is the concavity of the cross section along the black dotted line?

The “un-mixed” partials: fxx and fyyfxx(R) is

Positive

Negative

Zero

Example 3

What is the concavity of the cross section along the black dotted line?

The “un-mixed” partials: fxx and fyy

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.

Example 3

The mixed partials: fxy and fyxfxy(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?

The mixed partials: fxy and fyxfyx(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?

The mixed partials: fxy and fyxfxy(Q) is

Positive

Negative

Zero

Example 2

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 fyxfyx(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?

The mixed partials: fyx and fxyfxy(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?

The mixed partials: fyx and fxyExample 3 fyx(R) is

Positive

Negative

ZeroWhat 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 fxyExample 3 fyx(R) is

Positive

Negative

ZeroWhat 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. . .

Example 4 fyx(R) is

Positive

Negative

ZeroWhat 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?

W

To see this better. . .

Example 4

The “cross” slopes go from

Positive to negative

Negative to positive

Stay the same

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?

W

fyx(R) is

Positive Negative Zero

To see this better. . .

Example 4

W