Computing the Directional Derivative ....

[Math Amateur]

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

Help will be appreciated ...

Peter

 

[pasmith]

I agree; none of the examples given in 2.4(a) make sense as functions from $\mathbb{R}^3$ to $\mathbb{R}$.

Are there any hints or solutions in the text which shed light on what the author intended to say here?

 

[Jon Pierre]

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:View attachment 236656
My question/problem is as follows:

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

Help will be appreciated ...

Peter

Grrrrrr! Thank you for bringing this to my attention!

That question is a very bad typo. :( If I remember what I was doing when I wrote that question is that I mean for (i) to be f(x,y,z)=x (so that the the function only actually depended on the one variable x and not on y and z) and then to have the functions in (ii) and (iii) to be functions that depended on y and z respectively. This was meant to help you contrast between these types of functions. But sadly, it is a typo for the time being so you are quite right to be confused. What I had really wanted was something like this:

(i) f(x,y,z) = x
(ii) f(x,y,z)=y^2-y
(iii) f(x,y,z)=cos(z)

So even though they are functions of three variables they really only depend on one variable. Consider how each of the v's effect each of these types of functions. If there is one thing I have learned from writing this book it is how hard it is to catch typos! Sorry again.