Welcome to our community

Be a part of something great, join today!

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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
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:


Fortney - Question 2.4 .png


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
Jan 16, 2013
255
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
Jun 22, 2012
2,886
Hobart, Tasmania
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