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Gradient

Gradient

It is interesting to consider the rat of change for each direction of a vector in space. A particle could change its direction a different ways and different velocities. If you know its way in space you could predict where it would be then and then.

In mathematics we have a wonderful instrument to find out the rate of change by using the partial derivatives .

x, y and z are the three dimensions of the cartesian coordinates.

  Definition

The gradient is defined as the vector of partial derivatives:

The gradient gives us the rate of changes for all three components as a vector.

The gradient of the function f(x, y) = −(cos²x + cos²y)² depicted as a projected vector field on the bottom plane (Source: Simiprof)

  Example

Find the gradient of

Derive partially:

 is a constant with respect to x.

x is a constant with respect to y.

2 is a constant with respect to z.

The gradient of the vector is: