Vector calc proof question

This isn't homework. I was browsing through my old James Stewart multivariable calc textbook and am having a mental block concering an aspect of the "proof", shown in the file attachment, below. I've highlighted the portion giving me trouble.

My confusion concerns how the first integral (the line integral for C1) can be regarded as independent of x. The integral isn't independent of r which I assume would be dependent on both x and y.

http://home.comcast.net/~ut1880h/Files/stewart_theorem.jpg" [Broken]

jf

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tiny-tim
Homework Helper
My confusion concerns how the first integral (the line integral for C1) can be regarded as independent of x. The integral isn't independent of r which I assume would be dependent on both x and y.

Hi jackiefrost!

The integral is $$\int_{(a.b)}^{(x_1,y)}\bold{F}\cdot d\bold{r}$$

This is a function of x1 and y only.

If you increase or decrease x a little (keeping y the same), you can still use the same x1 … so, locally, x1 is independent of x (and so is y, of course).

Hi jackiefrost!
If you increase or decrease x a little (keeping y the same), you can still use the same x1 … so, locally, x1 is independent of x (and so is y, of course).
Hi tt,
Thanks for the insight. Once was I blind but now I see...

I'm starting to appreciate how cleverly this proof is structured to efficiently arrive at the desire end. It kinda tickles my head in a way that feels good (if ya know what I mean)

jf

tiny-tim
Homework Helper
thou once wast lost, but now art found …

Hi tt,
Thanks for the insight. Once was I blind but now I see...

I'm starting to appreciate how cleverly this proof is structured to efficiently arrive at the desire end. It kinda tickles my head in a way that feels good (if ya know what I mean)

jf

Hi jf,

it is a champagne amongst proofs!

hallelujah!

mathwonk