Proving Independence of x in Line Integral for Vector Calculus Proof

[jackiefrost]

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.

Thanks in advance...

Sorry - the file attachment was hard to read - here is a link instead...

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

jf

 

[tiny-tim]

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).

 

[jackiefrost]

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 you know what I mean)

jf

 

[tiny-tim]

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 you know what I mean)

jf

Hi jf,

it is a champagne amongst proofs!

hallelujah!

™

 

[mathwonk]

this is a trivial fact that follows instantly from the FTC. write your field as Pdx + Qdy, and then define the primitive function f(p) as the path integral of Pdx + Qdy taken along any path from a fixed point a to p.

all you need to do is show the x partial of f is P and the y partial is Q.

since the integral is the same for all paths, you choose the path to make this calculation easy.

e.g. to calculate ∂f/∂x make the path piecewise rectangular, with the last bit parallel to the x axis.

then the part of the integral for Qdy is zero (since y is not changing along a path parallel to the x axis), and the derivative of the Pdx bit is P by the FTC. etc...