Selected Topics: Directional Derivatives and The Gradient

This set of Mathematica graphics provides images of a directional derivative and the gradient.

We start with a surface, in this example , and a point .

This is a graph of the surface with the point.

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Then we zoom in on the surface near the point so we can concentrate on what's going on in that area.

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The next image shows the surface and the point, with the unit vector (in blue), a plane parallel to the vector intersecting the surface at and the tangent line at the point to the trace of the surface in the plane.

The rate of change of this line is called the "directional derivative of f at the point"

Now we consider all possible directional derivatives of f at that point. We ask the question: in which direction does f increase the fastest and what is the maximum rate of change?

Note that we are asking for a rate and a direction; such combinations are best described by vectors.

It turns out that a special vector, called the "gradient", is the vector that gives us the answers to our question above.

The gradient of a function f at a point (a,b) is given by:

We have a theorem that states that if f is a differentiable function of two variables, then the maximum value of the directional derivatives is the magnitude of the gradient vector and it occurs in the direction of the gradient vector.

The next image shows the surface and the point, with the gradient vector (in blue), a plane parallel to the gradient vector and containing the point and the tangent line at the point to the trace of the surface in the plane. The magnitude of the gradient vector gives the rate of change of this line.

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