Since we established in
our latest example that the maximal growth is somehow
related to the gradient, let us try to
generalize this concept for arbitrary function
and arbitrarily point. Firstly, let's start
with revisiting our calculation and growth for the directional derivative. Directional derivative equals to scalar product of the gradient at a given point and directional vector in case the directional
vector is normalized. Also we are going to
stress out one more time that we're particularly
interested into the geometrical sense of the scalar projects which
is the multiplication of the length of the vector with the cosine of the
angle in between them. So the question is; for a fixed point, fixed function what
is the direction of maximal growth,
that's stays forward. Moreover is if the
point is fixed, then one should expect that
partial derivative fixed. Thus we know the values at this very point as
the gradient is fixed and then simply known and
could not alter them. The lens of the direction is pretty much the
same which equals to one thus we couldn't
change it either. So as only thing we are
actually able to modify here is the cosine of the
angle in-between the gradient ends the direction
of the differentiation. So basically the only thing
changeable here is this. So by the definition of
the cosine function on the maximum value of the
cosine fraction is one, and it happens that
if the angle between two vectors equal to zero
or vectors are co-directed. So to sum up what it
does mean in terms of our maximal growth and the
direction of maximal growth, the speed of maximal growth. The maximum value of this directional
derivative for any point for every function is a
function is differentiable, is the length of the
gradient at this very point, and the direction
of maximal growth is the gradient itself. Basically, of course
formerly we should say is that the direction is the normalized vectors as we need not only stayed at its core directed with a gradient but we should normalize the gradient. But it doesn't change as
there are ideas that at this very point
the gradient shows not only the direction where to go to get
the maximal value, but how fast your function
changes in around there. So let us consider for
example some basic function like that equal to
X-squared plus Y-squared. So that's our parabola which has resulted after the
rotation of axons with state at which the first meeting of this factor in
the earliest weeks. So first let us start with considering as a gradient of
this function which is 2_X, 2_Y thus for any
possible point A, B. So direction of maximum
growth at this point, I'm going to write L max
is well basically 2A, 2B divided by the
length of this vector. It's complicated, and as the
value of the derivative on this direction is two square root of A squared plus B square. That was easier. By symmetric effects, the direction of
maximal growth is also state as a direction of minimal growth or
maximal decrease. If you will it is minus, gradient or inter gradient, and we're going to use it
in the following slide.