ANeTour.nb

Explicit Functions of Several Variables

In the Local Linearization section in this tour, we described the connection between tangent planes and total derivatives of a function

. The moral of that story was that we can apply the linear solution to a problem if we work at a small scale. One thing we want to compute is the rate of climb along a surface as we change the input quantities (or input vector)

in a certain direction.

The next figure shows the planar graph of a linear function , a horizontal disk through the plane and a circle of directions in the plane below the graph. One direction of change is shown in red. Notice that the gap between the disk and plane depends on the red direction of change. Mathematica can animate what happens as we vary the direction. Click on the link below the graph to run the animation.

Directional Rate of Change in the Linear Case

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Directional Rate of Change in the Nonlinear Case

The formula for the linear change in the previous linear problem is that the rate of change in direction vector

, where

is the gradient vector

. Local linearization lets us use this computation for the instantenous rate of change on a smooth surface, first finding

and

as partial derivatives evaluated at the starting point. The rate of change for a particular change vector

is shown in the next figure. Click on the link below the graph to run the Mathematica animation as we vary the direction.

[Graphics:../Images/index_gr_27.gif]

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Converted by Mathematica May 7, 2001