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Tony Stark
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The line element is defined as
How is dx2+dy2+dz2 be written as gijdqidqj.
Is some sort of notation used??
How is dx2+dy2+dz2 be written as gijdqidqj.
Is some sort of notation used??
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.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,
But how will we write the formula in Einsteins NotationMentz114 said:Yes. See here https://en.wikipedia.org/wiki/Einstein_notation
Like thisTony Stark said:But how will we write the formula in Einsteins Notation
Thanks Mentz. Is there an online link which has elaborate description of Einstein's Notation?? Please mentionMentz114 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.
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.Tony Stark said:Thanks Mentz. Is there an online link which has elaborate description of Einstein's Notation?? Please mention
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, ...),
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.
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.
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.
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.
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.