1. Second Order Partial Derivatives Curvature in Surfaces

2. The un-mixed partials: f xx and f yy We know that f x (P) measures the slope of the graph of f at the point P in the positive x direction. So f xx (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, f yy (P) measures the concavity of the graph along the y-cross section through P.

3. The un-mixed partials: f xx and f yy f xx (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: f xx and f yy f yy (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: f xx and f yy f xx (Q) is Positive Negative Zero What is the concavity of the cross section along the black dotted line? Example 2

6. The un-mixed partials: f xx and f yy f yy (Q) is Positive Negative Zero What is the concavity of the cross section along the black dotted line? Example 2

7. The un-mixed partials: f xx and f yy f xx (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: f xx and f yy f xx (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: f xx and f yy 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

10. The mixed partials: f xy and f yx f xy (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: f xy and f yx f yx (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: f xy and f yx f xy (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?

13. The un-mixed partials: f xy and f yx f yx (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: f yx and f xy f xy (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: f yx and f xy Example 3 f yx (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? ?

16. The mixed partials: f yx and f xy Example 3 f yx (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?

17. Sometimes it is easier to tell... Example 4 f yx (R) is 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? W

18. 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

19. fyx(R) is Positive Negative Zero To see this better... Example 4 W