# Line element in Euclidean Space

• Tony Stark
In summary, the line element in Einstein's notation is defined as ##dx^2+dy^2+dz^2=g_{ij}dx^i dx^j##, where the metric tensor gij is defined as gij = Ei*Ej. This is only true in a Cartesian coordinate system, but there are several possible coordinate systems on Euclidean space. The Einstein convention is that repeated indices are summed over all possible values. Online resources are available to explain this notation in more detail, but there may be some ambiguity in the conventions used.
Tony Stark
The line element is defined as

How is dx2+dy2+dz2 be written as gijdqidqj.
Is some sort of notation used??

The metric tensor gij is defined as gij = Ei*Ej, you can see that in Euclidean (flat) space that gij is to 0 whenever i is not equal to j but gij = 1 when i=j, now you may want to simplify the whole equation!

Noctisdark said:
The metric tensor gij is defined as gij = Ei*Ej, you can see that in Euclidean (flat) space that gij is to 0 whenever i is not equal to j but gij = 1 when i=j,
This is only true in a Cartesian coordinate system. There are several possible coordinate systems on Euclidean space which are neither orthogonal nor normalised. Generally, the metric tensor defines the inner product instead of the other way around.

PWiz and Noctisdark
Tony Stark said:
But how will we write the formula in Einsteins Notation
Like this
##dx^2+dy^2+dz^2=g_{ij}dx^i dx^j##

Remember that ##x^1\equiv x, x^2\equiv y, x^3 \equiv z##. ##i,j## are spatial indexes.

Tony Stark
Mentz114 said:
Like this
##dx^2+dy^2+dz^2=g_{ij}dx^i dx^j##

Remember that ##x^1\equiv x, x^2\equiv y, x^3 \equiv z##. ##i,j## are spatial indexes.
Thanks Mentz. Is there an online link which has elaborate description of Einstein's Notation?? Please mention

Tony Stark said:
Thanks Mentz. Is there an online link which has elaborate description of Einstein's Notation?? Please mention
That's all I got. You'll need to understand tensor notation. It is well explained in lots of books and online articles and courses.

What's wrong with the wiki link Mentz gave? Was it unclear? Too technical?

Basically, the Einstein convention is that you sum over repeated indices. In your example, ##g_{ij} dx^i dx^j## i and j are repeated, so you sum over i,j having all possible values. The Wiki link explains this - or tries to, so have several posters.

There is some ambiguity - what are all possible values? There are conventions for that too. But while textbooks will generally explain their conventions and follow them rigorously, you won't always see all posters here (including me) follow suit, leaving it up to the reader to sort out the minor ambiguities.

Wiki says:
In general relativity, a common convention is that

• the Greek alphabet is used for space and time components, where indices take values 0,1,2,3 (frequently used letters are μ, ν, ...),
• the Latin alphabet is used for spatial components only, where indices take values 1,2,3 (frequently used letters are i, j, ...),
So using the above convention ##g_{ij} dx^i dx^j## in cartesian coordinates, where by modern convention ##t= x^0 \quad x = x^1 \quad y = x^2 \quad z=x^3## would be just dx^2 + dy^2 + dz^2, while ##g_{\mu\nu} dx^\mu dx^\nu## would be the space-time interval. Depending on more sign conventions, the space-time interval might be interpreted either as dx^2 + dy^2 + dz^2 - dt^2 , or possibly dt^2 - dx^2 - dy^2 - dz^2.

In cylindrical coordinates, we might have coordinates ##r, \phi, z## rather than x,y,z. If we identify ##x^1=r \quad x^2 = \phi \quad x^3=z## (the exact assignment here may vary even more between writers) then we would still write ##g_{ij} dx^i dx^j##, but this corresponds to ##dr^2 + r^2d\phi^2 + dz^2##, which you will hopefully recognize as the formula for the line element for distance in cylindrical coordinates. So in cylindrical coordinates with the given assignments ##g_{11}=1 \quad g_{22}=r^2 \quad g_{33}=1##. The result, though, gives you the distance in whatever coordinate system you use.

There are a lot of ambiguities in the notation, either a fuller explanation is given (in a well-written textbook), or the reader is expected to fill in the details from context.

vanhees71

## 1. What is a line element in Euclidean Space?

A line element in Euclidean Space is a mathematical concept that defines the distance between two points in a straight line. It is represented by the symbol ds and is used in the study of geometry and calculus.

## 2. How is a line element calculated?

The line element is calculated using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the case of a line element, the hypotenuse represents the distance between two points and the other two sides represent the differences in their coordinates.

## 3. What is the significance of the line element in Euclidean Space?

The line element is an important concept in Euclidean Space as it allows for the calculation of distances between points and the measurement of lengths of curves. It also plays a crucial role in the study of geometry, as it is used to define properties such as parallelism, perpendicularity, and angles.

## 4. How does the line element differ from the arc length?

The line element and the arc length are two different ways of measuring distances in Euclidean Space. The line element measures the distance between two points in a straight line, while the arc length measures the distance along a curve between two points. The line element is a straight line, whereas the arc length may be curved.

## 5. Can the line element be extended to other spaces?

Yes, the concept of the line element can be extended to other spaces, such as non-Euclidean spaces. In these spaces, the line element may be defined using different mathematical equations, depending on the specific space being studied. However, the basic principle of measuring distance between two points remains the same.

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