Infinitesimal arc-length square

• andlook
In summary, the (ds)^2=(dx)^2+(dy)^2+(dz)^2 represents the infinitesimal arc-length square. It is used to calculate the arc length of a parameterized curve by taking the limit of polygonal approximations. The second metric, (ds)^2=(dx)^2+(1+x^2)(dy)^2 -2x(dy) +(dz)^2, is not a commonly used expression and its application is unclear.
andlook
Hello

I am trying to understand the "infinitesimal arc-length square." So (ds)^2=(dx)^2+(dy)^2+(dz)^2. What does this means?

And then what does (ds)^2=(dx)^2+(1+x^2)(dy)^2 -2x(dy) +(dz)^2 mean? And how does this apply to a space?

If you have a parameterized curve

$$\vec r(t) = \langle x(t), y(t), z(t) \rangle,\, t \in [a,b]$$

you calculate the arc length by taking the limit of polygonal approcimations. If

$$a = t_0 < t_1 < ...< t_n = b$$

the polygonal approximation to the arc length is

$$P_n = \sum_{i=1}^n |\vec r(t_i) - \vec r(t_{i-1})|$$

Here the absolute value signs represent Euclidean distance so

$$|\vec r(t_i) - \vec r(t_{i-1})| = \sqrt{(x(t_i)-x(t_{i-1}))^2+(y(t_i)-y(t_{i-1}))^2+(z(t_i)-z(t_{i-1}))^2}$$

This is sometimes written the the delta notation:

$$\Delta s_i \approx \Delta P_i =\sqrt{(\Delta x_i)^2 + (\Delta y_i)^2 + (\Delta z_i})^2$$

Passing to the limit suggests the notation $ds^2 = dx^2+dy^2+dz^2$

Thanks, that's great.

I'm not really sure if this question would be well defined, but how would I then apply this method to the second metric $$(ds)^2=(dx)^2+(1+x^2)(dy)^2 -2x(dy) +(dz)^2$$?

andlook said:
Thanks, that's great.

I'm not really sure if this question would be well defined, but how would I then apply this method to the second metric $$(ds)^2=(dx)^2+(1+x^2)(dy)^2 -2x(dy) +(dz)^2$$?

I can't say I have seen that kind of expression. The type of line integral you usually see in calculus is one of these two:

$$\int_C \vec F \cdot\, d\vec R\ or\ \int_C \delta(x,y,z)\ ds$$

The first might represent work in a force field and the second the mass of a wire. Perhaps you are thinking of some other setting or application.

1. What is an infinitesimal arc-length square?

An infinitesimal arc-length square is a concept in calculus that refers to a small, almost non-existent square formed by an infinitely small arc. It is used to calculate the length of a curve in a given interval.

2. How is an infinitesimal arc-length square calculated?

To calculate an infinitesimal arc-length square, the formula ds = √(dx² + dy²) is used, where ds represents the length of the arc, dx represents the change in the x-axis, and dy represents the change in the y-axis.

3. What is the significance of infinitesimal arc-length squares in mathematics?

Infinitesimal arc-length squares play a crucial role in calculus and are used to solve problems related to curves and their lengths. They also help in understanding the concept of limits and derivatives.

4. Can infinitesimal arc-length squares have a non-zero area?

No, by definition, infinitesimal arc-length squares have an infinitely small area. This means that they are so small that their area can be considered to be zero.

5. How are infinitesimal arc-length squares related to the concept of limits?

Infinitesimal arc-length squares are related to the concept of limits because they represent the smallest possible change in the independent variable (x) and the dependent variable (y). This helps in understanding the behavior of a function as the change in the variables approaches zero.

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