1 / 19

Second Order Partial Derivatives

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.

guy-pruitt
Download Presentation

Second Order Partial Derivatives

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Second Order Partial Derivatives Curvature in Surfaces

  2. 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.

  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 Example 2 fxx(Q) is Positive Negative Zero 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 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: 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? The “cross” slopes go from Positive to negative Negative to positive Stay the same Example 4 W

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

More Related