# Coordinate systems. Derivatives

• LagrangeEuler
In summary, the conversation discusses the use of unit vectors in cylindrical systems, specifically the equations for \vec{e}_{\rho} and \vec{e}_{\varphi}. The use of these equations is applicable for any orthogonal system, as dividing a vector by its magnitude results in a unit vector. The symbols \vec{r}, \rho, \varphi, \vec{i}, \vec{j}, and \vec{k} were not defined in the conversation.

#### LagrangeEuler

## \vec{r}=\rho \cos \varphi \vec{i}+\rho \sin \varphi \vec{j}+z\vec{k} ##
we get
$$\vec{e}_{\rho}=\frac{\frac{\partial \vec{r}}{\partial \rho}}{|\frac{\partial \vec{r}}{\partial \rho}|}$$
$$\vec{e}_{\varphi}=\frac{\frac{\partial \vec{r}}{\partial \varphi}}{|\frac{\partial \vec{r}}{\partial \varphi}|}$$
Is it correct for any orthogonal system or maybe for any system? Why you can use this relation?

It would help if you would define the symbols, $$\vec{e}_{\varphi}, \vec{e}_{\rho}$$

Unit vectors in cylindrical system.

If you divide any vector by its magnitude you get a unit vector. I am not sure what you are asking?

## 1. What is a coordinate system?

A coordinate system is a mathematical framework used to locate and describe points in space. It consists of a set of axes, usually labeled x, y, and z, and a reference point called the origin.

## 2. What are the different types of coordinate systems?

There are several types of coordinate systems, including Cartesian, polar, cylindrical, and spherical. Each type has its own unique way of defining coordinates and measuring distances.

## 3. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function with respect to its independent variable. It is commonly used in calculus to solve problems involving rates of change and optimization.

## 4. How do you calculate derivatives?

Derivatives can be calculated using various methods, such as the power rule, product rule, quotient rule, and chain rule. These methods involve taking the limit of a function as its independent variable approaches a specific value.

## 5. What are some real-world applications of coordinate systems and derivatives?

Coordinate systems and derivatives are used in a variety of fields, including physics, engineering, economics, and computer graphics. They are essential for modeling and analyzing motion, predicting future behavior, and creating accurate visual representations of objects and systems.