# Computing the Directional Defivative ... Fortney, Question 2.4 (a) (i) ... ...

#### Peter

##### Well-known member
MHB Site Helper
I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Chapter 2: An Introduction to Differential Forms ...

I need help with Question 2.4 (a) (i) concerned with computing a directional derivative ...

Question 2.4, including the preceding definition of a directional derivative, reads as follows:

My question/problem is as follows:

In question 2.4 (a) (i) we are asked to find $$\displaystyle v_p[f]$$ where $$\displaystyle f$$ is given as $$\displaystyle f(x) = x$$ ... .... BUT ... $$\displaystyle v$$ and $$\displaystyle p$$ are given in $$\displaystyle \mathbb{R}^3$$ ... so doesn't $$\displaystyle f$$ need to be defined on $$\displaystyle \mathbb{R}^3$$ ... say something like $$\displaystyle f(x,y, z) = x$$ or similar ...

Help will be appreciated ...

Peter

Last edited:

#### GJA

##### Well-known member
MHB Math Scholar
Hi Peter,

I agree with your analysis; the question does not appear to be well posed. My best guess is that the author simultaneously means $x$ represents a point in $\mathbb{R}^{3}$, as well as the $x$ coordinate of a point in $\mathbb{R}^{3}$ when using a Cartesian coordinate system; i.e., $x = (x, y, z)$. I am not familiar with Fortney's book, so this is merely a guess on my part.

#### Peter

##### Well-known member
MHB Site Helper
Hi Peter,

I agree with your analysis; the question does not appear to be well posed. My best guess is that the author simultaneously means $x$ represents a point in $\mathbb{R}^{3}$, as well as the $x$ coordinate of a point in $\mathbb{R}^{3}$ when using a Cartesian coordinate system; i.e., $x = (x, y, z)$. I am not familiar with Fortney's book, so this is merely a guess on my part.

Thanks GJA ...

Yes, I thought about $x = (x, y, z)$ ... but rejected the idea given that Fortney defines the directional derivative for function $$\displaystyle f: \mathbb{R}^n \to \mathbb{R}$$ ...

Thanks again ...

Peter