# I Directional Derivative demonstration

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1. Jan 1, 2018

### mech-eng

I find directional derivatives confusing. For example if there is a change in a direction and if this direction have both x and y components should not the change be calculated as square root of squares, i.e the pythogores theorem? Would you please provide a simple demonstration showing the directional derivative in terms of partial derivatives with respect to both x and y.

Thank you.

2. Jan 1, 2018

### PeroK

The directional derivative is the derivative in a given direction. Whatever that direction, you could imagine rotating your function so that the direction lies along the x-axis. In which case, the directional derivative is just the partial derivative with respect to x.

Otherwise, the directional derivative is just the relevant proportions of the various partial derivatives. For example if the unit vector in the given direction is $\vec{u} = (a, b)$, then:

$D_{\vec{u}}f = a f_x + bf_y$

There's a proof of that here:

http://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx

Does that help?

3. Jan 1, 2018

### mech-eng

That's not much help. I hope to see a visual example. How we demonstrate decomposition of a vector in a plane by projecting its components onto x and y-axes. Then we put one's tail to other's head to sum them. That example is easy to demonstrate. But how can we do this with directional derivatives?

Thank you.

4. Jan 1, 2018

### mech-eng

This is confusing for me because there is a magnitude in x direction and there is a magnitude in y-direction. These magnitudes are magnitudes of rate of changes. These two directions are perpendicular to each other. So should not the Pythaogoras theorem be applied?

5. Jan 1, 2018

### PeroK

"Confusing" or "surprising"? The proof shows why $a$ and $b$ come out, rather than $\sqrt{a^2 + b^2}$. You can't just apply Pythagoras. This is about the rate of change of a function, not the magnitude of a vector with two components.

6. Jan 1, 2018

### mech-eng

But isn't there any geometric explanation or interpretation or even an animation?

Thank you.

7. Jan 1, 2018

### PeroK

Well, I guess you can google for an animation. I can't immediately see a neat geometric explanation - not that that means there isn't one. The proof comes from the chain rule for derivatives, rather than Pythagoras.

Interesting!

8. Jan 1, 2018

### mech-eng

Yes, I had started before you wrote this and I encountered with this. This is a strange tool. Have you ever used before? A Computable Document does not make any sense to me.

One of the important key phrase is "Visualize directional derivatives".