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Second Order Partial Derivatives. Curvature in Surfaces. We know that f x ( P ) measures the slope of the graph of f at the point P in the positive x direction.

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the un mixed partials f xx and f yy
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.

The “un-mixed” partials: fxx and fyy

Likewise, fyy(P) measures the concavity of the graph along the y-cross section through P.

the un mixed partials f xx and f yy1
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?

the un mixed partials f xx and f yy2
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?

the un mixed partials f xx and f yy3
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 f xx and f yy4
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?

the un mixed partials f xx and f yy5
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?

the un mixed partials f xx and f yy6
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?

the un mixed partials f xx and f yy7
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 f xy and f yx
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?

the mixed partials f xy and f yx1
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?

the mixed partials f xy and f yx2
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 f xy and f yx
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?

the mixed partials f yx and f xy
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?

the mixed partials f yx and f xy1
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?

the mixed partials f yx and f xy2
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?

sometimes it is easier to tell
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?

to see this better
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?

The “cross” slopes go from

Positive to negative

Negative to positive

Stay the same

Example 4

W

to see this better1
To see this better. . .
  • fyx(R) is
  • Positive
  • Negative
  • Zero

Example 4

W

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