is the normal vector. It can be pointing in or out of the sphere, whereas points out of the sphere.
I've run into a case, where one should calculate the directional derivative of a function of several variables at a point on a sphere of radius in the direction of a normal vector to this sphere. It is known that the function itself can be written only in terms of the length of the argument x, i.e. .
Why is it actually allowed to write , with for the outer/inner normal vector?
I think there is a misunderstanding here. By a function of several variables I mean explicitly
is the euclidean length of .
I thought this was clear. Anyway, it's my fault. Sorry.
The notation is somewhat arbitrary as long it is clear from the context what it means.
Since you said it was a directional derivative in the direction normal to the sphere, I would interpret your derivatives to mean:
Although it may also be possible that your vectors need to be normalized, depending on how your book defines a directional derivative.
See Directional derivative - Wikipedia, the free encyclopedia for more information on how a directional derivative is or can be defined.
Either way, those 2 forms are the same, except for a possible minus sign.