How To Find The Vector With Maximum Steepness Using Directional Derivatives And Gradient Vectors In Math Functions

Find vector with maximum steepness

The gradient vector at that point

To find the vector with maximum steepness, we need to first define what we mean by steepness. One possible measure of steepness is the magnitude of the directional derivative in the direction of the vector.

Let f(x,y) be a function of two variables. The directional derivative of f at a point (x0, y0) in the direction of a unit vector u = is given by:

df/ds = ∇f(x0, y0)·u

where ∇f(x0, y0) is the gradient of f at (x0, y0).

The magnitude of the directional derivative in the direction of u is given by |df/ds|. To maximize this quantity, we need to choose u such that ∇f(x0, y0)·u is as large as possible in absolute value.

One way to do this is to choose u to be the unit vector in the direction of ∇f(x0, y0). This is because the dot product of two unit vectors is equal to the cosine of the angle between them. Therefore, the largest possible value of ∇f(x0, y0)·u is equal to the magnitude of ∇f(x0, y0).

In other words, the vector with maximum steepness is the gradient vector of the function at the given point, and its magnitude gives the rate of change of the function in that direction. The direction of the gradient vector also points towards the direction of the greatest increase of the function at the given point.

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