Second Order Partial Derivatives Curvature in Surfaces The

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Second Order Partial Derivatives Curvature in Surfaces

Second Order Partial Derivatives Curvature in Surfaces

The “un-mixed” partials: fxx and fyy We know that fx(P) measures the slope of

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 fyy Example 1 fxx(P) is Positive Negative Zero What

The “un-mixed” partials: fxx and fyy Example 1 fxx(P) is Positive Negative Zero What is the concavity of the cross section along the black dotted line?

The “un-mixed” partials: fxx and fyy Example 1 fyy(P) is Positive Negative Zero What

The “un-mixed” partials: fxx and fyy Example 1 fyy(P) is Positive Negative Zero What is the concavity of the cross section along the black dotted line?

The “un-mixed” partials: fxx and fyy Example 2 fxx(Q) is Positive Negative Zero What

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: fxx and fyy Example 2 fyy(Q) is Positive Negative Zero What

The “un-mixed” partials: fxx and fyy Example 2 fyy(Q) is Positive Negative Zero What is the concavity of the cross section along the black dotted line?

The “un-mixed” partials: fxx and fyy Example 3 fxx(R) is Positive Negative Zero What

The “un-mixed” partials: fxx and fyy Example 3 fxx(R) is Positive Negative Zero What is the concavity of the cross section along the black dotted line?

The “un-mixed” partials: fxx and fyy Example 3 fxx(R) is Positive Negative Zero What

The “un-mixed” partials: fxx and fyy Example 3 fxx(R) is Positive Negative Zero What is the concavity of the cross section along the black dotted line?

The “un-mixed” partials: fxx and fyy Example 3 Note: The surface is concave up

The “un-mixed” partials: fxx and fyy Example 3 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.

The mixed partials: fxy and fyx Example 1 fxy(P) is Positive Negative Zero What

The mixed partials: fxy and fyx Example 1 fxy(P) is Positive Negative Zero 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 fyx Example 1 fyx(P) is Positive Negative Zero What

The mixed partials: fxy and fyx Example 1 fyx(P) is Positive Negative Zero 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 fyx Example 2 fxy(Q) is Positive Negative Zero What

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: fxy and fyx Example 2 fyx(Q) is Positive Negative Zero What

The “un-mixed” partials: fxy and fyx Example 2 fyx(Q) is Positive Negative Zero 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 fxy Example 3 fxy(R) is Positive Negative Zero What

The mixed partials: fyx and fxy Example 3 fxy(R) is Positive Negative Zero 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 fxy Example 3 fyx(R) is ? Positive Negative Zero

The mixed partials: fyx and fxy Example 3 fyx(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?

The mixed partials: fyx and fxy Example 3 fyx(R) is Positive Negative Zero What

The mixed partials: fyx and fxy Example 3 fyx(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?

Sometimes it is easier to tell. . . fyx(R) is Example 4 W Positive

Sometimes it is easier to tell. . . fyx(R) is Example 4 W 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?

To see this better. . . What happens to the slope in the y

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? Example 4 W The “cross” slopes go from Positive to negative Negative to positive Stay the same

To see this better. . . n Example 4 W n n n fyx(R)

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

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