- #1
Swimmingly!
- 44
- 0
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 of r.
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
[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 of r.
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