Hi,
At page 187, is introduced:

But I'm not sure what is meant by this.
Somewhere is written that but I simply don't understand this section. For example, doesn't return a vector, right? Why is then called a 'vector field'?
Furthermore, I still wonder what is 'dx' and the '1-form'. I think, I miss anything to understand this section.
Does anyone know how it works?
Greetings,
fruity

You may or may not find it useful to read this discussion in the forum:
viewtopic.php?f=4&t=1439

here is an operator, so

can be thought of as a vector since it is the derivative in the direction of the x axis.

Thanks... I will have to think about it (,too).

Ok, I what I've understood now:
is the direction we want to calculate the slope in.
is the covector to the vector along the contour, but I'm not sure. There is the equation , which says me that the slope vanishes when the vector points in the right direction (along the contour). also says me that has to be the covector of , right? I think I still miss something related to ...


So this equation in the book didn't mean the vector but the operator or , which returns a number...
Furthermore, is called a vector field. Is now an operator or a vector or a vector field?

is not a vector, but a scalar field on S (see page 183 section 10.2 near the top), so you can't write .
is also not a vector.

Sorry, I meant

and

. What I still don't understand is why should "give the direction of the contours" (Fig. 10.8). Isn't it the same as ?

Hi fruity
By the way I don't understand this topic either, I'm just hoping that by discussing it with someone else I might get there.

Thanks for your help. I think what I know now is enough for reading the next chapters...
Maybe the use of these things in later chapters will help, too.