What is the meaning of the directional derivative of a function mean?

[Outrageous]

eg. Find the directional derivative of the function phi=xyz^2 at the point (1,2,3).
Actually what is the math used for?
Let's say
phi is the temperature of air(scalar field).
∇phi will be the rate if change of temperature at (1,2,3), why the direction come out.
directional derivative of it is the velocity of air(vector field)?
If so , can I say directional derivative of it is the direction of the air?

actually this is the conclusion I made from my own study.
please give me a simple example.
Thank you

 

[pellman]

It is the rate of change of a function in a given direction. This is trivial for a function of one variable since there is only one direction. With multiple variables, you can't simply ask "what is the rate of change of the function?" because at any given point it may change at different rates in any of the infinite directions you can take from that point. You have to pick a direction and ask "what is the rate of change in that direction?"

For one 1 variable we have

$$df/dx = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$

For multiple variables your small increment has to be in some particular direction. So we write for the directional derivative in the direction of v

$$\lim_{h\rightarrow 0} \frac{f(\textbf{x}+h\textbf{v})-f(x)}{h|\textbf{v}|}$$

The answer turns out to be $\nabla f \cdot \textbf{v}/|\textbf{v}|$

 

[pellman]

If so , can I say directional derivative of it is the direction of the air?

The direction of the directional derivative is arbitrary. There may be some physical reason why you choose a particular direction. For instance you might want to know the rate of change of temperature in the direction of the air flow. But mathematically you are free to take the directional derivative of the temperature in any direction.

Another example, when a symmetry is present you may want the directional derivative in a meaningful direction. In geology, you might want the rate of change of temperature as you move toward the center of the earth.

 

[Outrageous]

With multiple variables, you can't simply ask "what is the rate of change of the function?" because at any given point it may change at different rates in any of the infinite directions you can take from that point. You have to pick a direction and ask "what is the rate of change in that direction?"

Thank you.
To double confirm.
So if my psi is temperature, and I want to find the rate of change of temperature in a direction, v , then I have to use the directional derivative of the psi (dot product) the unit vector of v.
The directional derivative of the psi is the rate of change of temperature in any direction in multivariable (if the direction is not given)
Is that how we apply this math on problem?

 

[Outrageous]

phi=function of (x,y,z) at point (1,3,2)
then I get ∇phi =-2i+13j+k , what is the physical meaning of it?
it is the rate of change of temperature in any direction.
You mentioned:
With multiple variables, you can't simply ask "what is the rate of change of the function?" because at any given point it may change at different rates in any of the infinite directions you can take from that point.

but why ∇phi here has a direction?

 

[pellman]

but why ∇phi here has a direction?

$\nabla\phi$ is not the directional derivative. $\nabla\phi$ is the gradient. $\nabla\phi\cdot\textbf{v}$ (where v is a unit vector) is the directional derivative in the direction of v.

The gradient of a function is a vector which points in the direction of the most rapid rate of change of the function. and the value of the gradient is that rate of change. That is, the maximum value of the directional derivative $\nabla\phi\cdot\textbf{v}$ is when

$\textbf{v}=\frac{\nabla\phi}{|\nabla\phi|}$

 

[Outrageous]

The gradient of a function is a vector which points in the direction of the most rapid rate of change of the function. and the value of the gradient is that rate of change.
$\textbf{v}=\frac{\nabla\phi}{|\nabla\phi|}$

Thank you .