Question

New Tutorial!!

Answer 1

here is my tutorial on the Euler Theorem

Answer 2

Great job! :)

Answer 3

thats awesome

Answer 4

thanks! @TheSmartOne @.Gjallarhorn. :)

Answer 5

Great Work!!

Answer 6

thanks! :) @Aureyliant

Answer 7

bad

Answer 8

I'm still reading through it, but I wanted to check if for \(\alpha\) as the grade of the homogeneous function, do you know (using Einstein summation notation) if there's a way to write this final step in my attempt at a generalization:

\[\frac{\partial f}{\partial x^i} x^i = \alpha f\]
\[\frac{\partial^2 f}{\partial x^i \partial x^j} x^ix^j = \alpha (\alpha -1) f\]

So here the n represents number of indices, so n=1 and n=2 are shown above, but I guess I need to index my indices now for this to work... I was wondering if you knew how to write it properly or if I should be using the \(D_if\) or \(f_{,i}\) notation for denoting partial derivatives haha.
\[\frac{\partial^n f}{(\partial x^i)^n} (x^i)^n = \frac{\alpha!}{(\alpha - n) !} f\]

Great guide I have no experience with homogeneous equations before, and this was a nice introduction! I am not very experienced with tensors, so I was wondering if there's a specific reason why you use directional derivatives instead of finding a gradient and then taking the dot product with a direction. Or maybe you did and I missed it.

Answer 9

I think that \[\alpha \] is a natural number, namely:
\[\alpha \in \mathbb{N}\]
Your equations are correct, namely the application of Einstein's notation is correct. Since Einstein introduced his notation when he was developing the General Theory of Relativity, wherein there are complex tensor formulas, I think that the usage of that notation is it is unnecessary in the case of my tutorial. In general, when I have to write my scientific thought, I prefer to avoid complex or unecessary formulas if I can.
The partial derivative with respect to a direction, as I explained in my tutorial, is a more general concept, than the partial derivative with respect to a single variable. That is why, I preferred to introduce the directional derivative.
Finally I'd like to thank you for your appreciation to my work :)
@Empty

Answer 10

very informative

Answer 11

thanks! @welshfella

Answer 12

awesome job

Answer 13

thanks! @freegirl11112

Answer 14

I really love all the tutorials you make @Michele_Laino ! Keep up the good work!

Answer 15

Thanks! @Zale101