# Connection between cesaro equation and polar coordinates

• Jhenrique
In summary, the article discusses the relationship between the cartesian and polar tangential angles and how to express a curve in terms of r and θ using the curvature κ. Euler's solution connects the cartesian system (x, y) with the "cesaro system" (s, κ), but the polar tangential angle ψ* is also defined in the 2D plane. The derivative of ψ with respect to s is equal to κ, allowing for a connection between κ and the polar system. The article also provides an equation for the curvature vector in terms of the parameterized r(s) and θ(s).
Jhenrique
First, I'd like that you read this littler article (http://mathworld.wolfram.com/NaturalEquation.html). The solution given by Euler that coonects the system cartesian (x, y) with the curvature κ of the "cesaro system" (s, κ), is that the derivative of the cartesian tangential angle φ* wrt arc length s results the curvature κ.

However, in the 2D plane is definied the polar tangential angle ψ* too. Thus if I want express a curve given s and κ in terms of r and θ (or vice-versa) I need to establish a connection between the curvature κ and the polar system. So, would be correct to say that the derivative of ψ wrt s is equal to κ?

*
https://en.wikipedia.org/wiki/Tangential_angle
https://en.wikipedia.org/wiki/Subtangent
https://en.wikipedia.org/wiki/List_...rtesian_coordinates_from_Ces.C3.A0ro_equation

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In polar coordinates, the unit tangent to a curve that is parameterized as r(s) and θ(s) is given by:

$$\vec{u}=\frac{dr}{ds}\vec{i}_r+r\frac{dθ}{ds}\vec{i}_θ$$

So, the curvature vector is given by$$κ\vec{n}=\frac{d\vec{u}}{ds}=\left(\frac{d^2r}{ds^2}-r(\frac{dθ}{ds})^2\right)\vec{i}_r+\left(r\frac{d^2θ}{ds^2}+2\frac{dr}{ds}\frac{dθ}{ds}\right)\vec{i}_θ$$
where $\vec{n}$ is the unit normal to the curve.

Jhenrique said:
I must have misunderstood your question. I thought you were just interested in getting an equation for the curvature in terms of the parameterized r(s) and θ(s). Apparently not.

Chet

Thank you for sharing this interesting article and your question about the connection between the Cesaro equation and polar coordinates. I can provide a response to your inquiry.

The Cesaro equation, also known as the natural equation, is a mathematical tool used to describe curves in terms of their curvature. This equation connects the Cartesian coordinates (x, y) with the curvature κ of the curve in the Cesaro system (s, κ). In other words, it provides a way to express the curvature of a curve in terms of its position along the curve.

On the other hand, polar coordinates are a different coordinate system that uses the distance from the origin (r) and the angle from a fixed reference direction (θ) to describe a point in a 2D plane. In this system, the tangential angle ψ* is defined as the angle between the radius vector and the tangent to the curve at that point.

Now, to answer your question, it is not correct to say that the derivative of ψ wrt s is equal to κ. This is because ψ and κ are not directly related in the polar coordinate system. However, it is possible to establish a connection between the two by using the Cesaro equation. The derivative of ψ wrt s can be expressed in terms of κ using the Cesaro equation, as shown in the article you shared.

In summary, the Cesaro equation and polar coordinates are two different mathematical tools used to describe curves. While the Cesaro equation connects the Cartesian coordinates with the curvature of a curve, polar coordinates use the distance and angle to describe a point. While there is no direct connection between the two, the Cesaro equation can be used to establish a connection between the curvature and tangential angle in the polar coordinate system. I hope this helps clarify the relationship between these two concepts.

## 1. What is the Cesaro equation and how is it related to polar coordinates?

The Cesaro equation is a mathematical expression used to calculate the sum of a series. It is related to polar coordinates through the use of the polar coordinate transformation, which converts Cartesian coordinates (x,y) to polar coordinates (r,θ). This transformation is used in the Cesaro equation to calculate the sum of a series in polar coordinates rather than Cartesian coordinates.

## 2. How is the Cesaro equation used in real-world applications?

The Cesaro equation is commonly used in engineering and physics to calculate the average behavior of a system over time. It is also used in probability and statistics to determine the expected value of a random variable. Additionally, the Cesaro equation has applications in signal processing and numerical analysis.

## 3. What are the advantages of using polar coordinates in the Cesaro equation?

Using polar coordinates in the Cesaro equation can simplify complex calculations and make certain problems easier to solve. It also allows for a more intuitive understanding of the behavior of a system in terms of direction and magnitude, rather than just Cartesian coordinates.

## 4. Can the Cesaro equation be applied to any series?

Yes, the Cesaro equation can be applied to any series as long as it meets the necessary conditions for convergence. This includes both infinite and finite series.

## 5. How does the Cesaro equation relate to other mathematical concepts?

The Cesaro equation is closely related to other mathematical concepts such as the Riemann integral, Fourier series, and Taylor series. It is also connected to the concept of convergence and the study of infinite series in mathematics.

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