(adsbygoogle = window.adsbygoogle || []).push({}); Integral through a path in 2D (or ND) What's the usual "definition"?

[Bold letters are vectors. eg:r]

We have a scalar function f(r) and a path g(x)=y.

I see two ways to reason:

(1) The little infinitesimals are summed with the change of x and on the change of y separately.

(2) The little infinitesimals are summed with the change ofr.

For example:

The scalar function is f(r)=1

The path is the straight line x=y, from x=0 to x=1.

(1) ∫dx+∫dy=1+1=2 ∫dx from 0 to 1, and since x=y, ∫dy from 0 to 1.

(2) ∫dr=√2 It's a straight path so ∫dr from 0 to √2.

What is the regular way to take an integral through a path?

(1) treats x and y totally independently, (2) seems more "physical/relative" but harder to calculate

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# Integral through a path in 2D (or ND) What's the usual definition ?

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