Gradient, Gradient vector, Directional derivative, Surface, Calculate the directional derivative.
THE GRADIENT AND THE DIRECTIONAL DERIVATIVE
The vector made up of the of the function's partial derivatives with respect to x and y is called the gradient of the function.
Here is the gradient:
, shortly .
The gradient helps us calculate the directional derivative. The directional derivative describes how steeply the surface of the function slopes along a given arbitrary direction.
So, it means that there is a mountain climber standing at point P on the surface, and decides to move in direction. The directional derivative tells him how steep he would have to climb.
Calculating the directional derivative is very simple: it is the dot product of the gradient and the unit-length vector .
The directional derivative of the function at point is:
( is a unit vector here)
Let's see an example of this!
Let's calculate the directional derivative of for direction at point .
According to the formula, the directional derivative is:
Here this funny symbol is the symbol of differentiation, and it is pronounced as "d", but there is a bit more friendly notation for the directional derivative: .
We need the partial derivatives for calculating the gradient.
So, the gradient is:
To get the directional derivative, we should create the dot product of the gradient and the vector, but it is now not a unit vector, its length is:
To turn this into a unit vector we divide the vector by its own length:
Now, let's see the vector.
The vector in the formula must be of unit length.
Since now is not a unit long,
we turn this into a unit vector.
We divide the vector by its own length:
Therefore the directional derivative is:
If a mountain climber asked us which direction he should take from point P in order to climb the steepest route, well...
we could actually give him an answer.
The steepest rise on a surface is always in the direction of the gradient vector.
That means if the climber starts climbing
in the direction of the gradient, then he will be climbing the steepest route.
The function is an explicit function, its derivative, as expected, is .