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The line element is defined as
How is dx2+dy2+dz2 be written as g_{ij}dq^{i}dq^{j}.
Is some sort of notation used??
How is dx2+dy2+dz2 be written as g_{ij}dq^{i}dq^{j}.
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.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 Notation
Like thisBut 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 mentionLike 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.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, ...),