Directional derivatives vs Partial derivatives

[Amaelle]

Homework Statement

The difference between partial and directional derivatives

Relevant Equations

directional derivatives

Good day
I just want to confirm if a function f(x,y) who has directional derivatives has automatically partial derivatives (even though the function itself is not necessarly differentiable)? Can we consider that partial derivatives are special cases of directional derivatives?
Thank you in advance!

 

[FactChecker]

The definitions of the directional derivative and of the partial derivative can be put in the "Relevant Equations" section.
You should look at the definition of the directional derivative and see if there are certain directions that would give the partial derivatives.

 

[haruspex]

Homework Statement:: The difference between partial and directional derivatives
Relevant Equations:: directional derivatives

Can we consider that partial derivatives are special cases of directional derivatives?

Yes, they are the directional derivatives in the directions of the "coordinates " (i.e. the parameters).

 

[Gaussian97]

If I remember correctly, I would say yes, that partial derivatives are special cases of directional derivatives (at least if the directional derivatives are defined with respect to vectors with norm 1).
Therefore if any directional derivative is defined for a function, the partial derivatives will be defined as well. But that doesn't mean that you can't find a function with directional derivatives for some directions, that don't have partial derivatives.
Therefore I would say that the implication is just the other way: If a function has partial derivatives, it will have directional derivatives (since partial derivatives are directional derivatives). But not the opposite, that is what you said.

 

[fresh_42]

Homework Statement:: The difference between partial and directional derivatives
Relevant Equations:: directional derivatives

Good day
I just want to confirm if a function f(x,y) who has directional derivatives has automatically partial derivatives (even though the function itself is not necessarly differentiable)? Can we consider that partial derivatives are special cases of directional derivatives?
Thank you in advance!

Yes. A derivative is always a directional derivative, which includes partial derivatives. Constructions like gradient, Jacobi matrices, or total derivatives are only collections of various directional derivatives, e.g. in form of a linear combination of partial derivatives.

Have a read: https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/

 

[Amaelle]

thank you very much
I just have another question
when a function admits directional derivatives, it means it admit it for all directions or maybe be there might exist function that admits directional derivatives in some directions but not in other directions?
thank you!

 

[Gaussian97]

If I remember correctly (I studied many years ago) there should be no problem to find a function that has directional derivative for some directions, but not for others.
If then such a function is said to have "directional derivatives" is something I'm not sure about. But it's just a nomenclature issue...

 

[fresh_42]

thank you very much
I just have another question
when a function admits directional derivatives, it means it admit it for all directions

This is usually meant if no specific direction is named. In such cases, it is an all quantifier.

or maybe be there might exist function that admits directional derivatives in some directions but not in other directions?
thank you!

There are such functions, e.g. if you imagine a curved three-dimensional $\mathcal{V}$. But we cannot say that such a function is directional differentiable because it is not in all directions. The link above contains some examples.