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.

 

[tacman]

The integral of G(x,y)dx, G(x,y)dy, and G(x,y)ds are all different types of integrals that represent different concepts in mathematics and physics. The choice of which integral to use depends on the specific problem or situation at hand.

The integral of G(x,y)dx represents the area under the curve of a function G(x,y) in the x-direction. This is commonly known as a double integral and is used to find the volume under a surface or the area between two curves. In this case, the independent variable is x and the integral is taken with respect to x.

On the other hand, the integral of G(x,y)dy represents the area under the curve of a function G(x,y) in the y-direction. This is also a double integral, but the independent variable is y and the integral is taken with respect to y. This type of integral is useful when finding the volume between two surfaces or the area enclosed by a curve.

Lastly, the integral of G(x,y)ds represents the arc length of a curve. In this case, the independent variable is the distance along the curve, represented by ds. This type of integral is commonly used in physics to find the work done by a force along a curved path or the total distance traveled along a curved path.

So, the difference between these integrals lies in what they represent and the independent variable used. The choice of which integral to use depends on the specific problem or situation at hand. For example, if you are trying to find the volume under a surface, you would use the integral of G(x,y)dx or G(x,y)dy, depending on whether the surface is bounded by curves in the x or y direction. If you are trying to find the work done along a curved path, you would use the integral of G(x,y)ds.

In summary, the integral of G(x,y)dx, G(x,y)dy, and G(x,y)ds represent different concepts and are used in different situations. The choice of which integral to use depends on the problem at hand and the independent variable being considered.