# Convert tensor from cartesian to cylindrical coordinate

• ngkamsengpeter
In summary, the conversation discusses the conversion of a tensor in terms of electromagnetic fields in Cartesian coordinates to cylindrical coordinates. The transformation is attempted using a transformation matrix and tensor transformation rule, but it does not yield the desired result. Further assistance is requested in solving the problem.
ngkamsengpeter

## Homework Statement

Given the tensor
$$F_{\mu \nu }= \left[ \begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & B_{z} &-B_{y} \\E_{y} & -B_{z} & 0 & B_{x} \\E_{z} & B_{y} & -B_{x} & 0 \end{array} \right]$$
$$F^{\mu \nu }F_{\mu \nu }=2(B^2-\frac{E^2}{c^2})$$
and metric tensor
$$n_{\mu \nu }= \left[ \begin{array}{cccc}c^2& 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right]$$

How to convert it into cylindrical coordinates, that is in terms of Eθ,Ez,Er

## The Attempt at a Solution

I try to convert it using the transformation matrix and tensor transformation rule but it turns out that
$$F^{\mu \nu }F_{\mu \nu }≠2(B^2-\frac{E^2}{c^2})$$

Can anyone give me some idea how to solve this?
Thanks.

No one can help?

## 1. What is a tensor?

A tensor is a mathematical object that describes the relationship between different coordinate systems. It is a multilinear mapping that associates a set of vectors, scalars, or tensors with a coordinate system.

## 2. What are cartesian coordinates?

Cartesian coordinates are a system of coordinates in which a point is located by its distance from a fixed origin along a set of mutually perpendicular axes. It is commonly used in three-dimensional space and is represented by the coordinates (x,y,z).

## 3. What are cylindrical coordinates?

Cylindrical coordinates are a system of coordinates that represent a point in three-dimensional space by its distance from a fixed origin, its angle from a fixed reference direction, and its height from a fixed plane. It is commonly used in cylindrical-shaped objects and is represented by the coordinates (r,θ,z).

## 4. Why convert from cartesian to cylindrical coordinates?

Converting from cartesian to cylindrical coordinates allows us to better understand the relationships between different coordinate systems and simplifies calculations for certain problems. Cylindrical coordinates are also useful when dealing with cylindrical-shaped objects or problems with rotational symmetry.

## 5. How do you convert a tensor from cartesian to cylindrical coordinates?

To convert a tensor from cartesian to cylindrical coordinates, we can use transformation rules that take into account the differences in the coordinate systems. These rules involve converting the basis vectors and components of the tensor from one system to the other. The resulting tensor will have the same physical meaning, but will be represented in cylindrical coordinates instead of cartesian coordinates.

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