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Second Order Partial Derivatives

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This article explores the concept of second-order partial derivatives, focusing on curvature in surfaces. We analyze how the "un-mixed" partial derivatives, fxx and fyy, indicate concavity along the x and y cross-sections. By examining different points (P, Q, R) on the surface, we illustrate how these derivatives provide insight into the surface's behavior in various directions. Additionally, we discuss mixed partial derivatives, fxy and fyx, to understand the change in slope around specified points. A comprehensive approach is taken to assess concavity and curvature across multiple examples.

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Second Order Partial Derivatives

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  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

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