Integral of G(x,y): dx vs. dy vs. ds

[LinearAlgebra]

If you have a curve integral, what is the conceptual or physical difference between integrating G(x,y)dx, G(x,y)dy and G(x,y)ds ?? How do you know when to do either one??

Confused. :shy:

 

[arunma]

If you have a curve integral, what is the conceptual or physical difference between integrating G(x,y)dx, G(x,y)dy and G(x,y)ds ?? How do you know when to do either one??

Confused. :shy:

You sound like you might be asking what the physical interpretations of these various line integrals are. The last one, G(x,y)ds, is called the line integral with respect to arc length. In single variable calculus, we say that if f(x) is positive, then the definite integral of f(x) over [a,b] represents the area under the graph of f(x) from a to b. In multivariable calculus, imagine graphing G(x,y) in an xyz space, and imagine the path C on the xy plane. We say that if G(x,y) is positive along path C, then the line integral of G(x,y)ds represents the area of the "curtain" that falls from G(x,y) down to path C on the xy plane.

There isn't any obvious physical description of the line integrals of G(x,y) with respect to x or y (not one that I can think of, anyway). But I can say this much. The differential dx represents a change from (x,y) to a new point, in the x direction, and dy represents an analogous small change in the y direction. The differential ds represents a change beginning at some point (x,y) and ending at another point, moving along curve C.

Computationally, these line integrals are very simple. If curve C can be parametrized according to two functions x(t) and y(t), then just rewrite the integrands like this:

G(x(t),y(t))\frac{dx}{dt}dt

G(x(t),y(t))\frac{dy}{dt}dt

G(x(t),y(t))\sqrt{(\dfrac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}}

And that's how you do line integrals.