![]() ![]() One can define Levi-civita symbol with n index as followsĪgain as the same way before we can equivalently express this in terms of determinant, thus the two product of them are nothing but Generalization of n indices of Levi-civita symbol ![]() We are focusing on Levi-Civita symbol which has three index with totally anti-symmetric, thus we haveĪnd this plays a important rule for derivation of vector calculus identities. Symmetric symbol S_[ij} and anti-symmetric symbol A_ are defined byįrom this the contraction of symmetric symbol and anti-symmetric symbol always vanishes. Note you can see the three indices i,j,k are totally-anti symmetric.įrom this we can easily show some identities related with cross product About symmetry The other equivalent definition of Levi-civita symbols are The cross product between two vectors is nothing but a function whose input is two vectors and output is a vector, we can write them as follows we can write them as followsĪs from the definition, kronecker delta symbol is symmetric under the exchange of i and j Cross product and Levi-civita symbol The inner product between two vectors is nothing but a function whose input is two vectors and output is a scalar. This notation is nothing to do with math but for saving some papers. Thus for the expression of repeated index, you should note that for your computation summation process for that indices is required. Einstein summation is nothing but a omit the sigma notation for summation as a convenience. Here since we are assuming flat, the upper index and lower index does not give big difference. I'll skip some basic explanation on what is vector, but starting how to express vectors. And then after, at the second post, we will do explicit computation on these equations. To do so we need some background, So at this post first we will cover Einstein summation convention and representation of inner product and cross product. One can prove this with explicit computation via coordinate but here using some symbolic we will prove these equations. These kinds of vector identities appeared in vector calculus, mathematical physics, mechanics, electrodynamics, fluid mechanics and many engineering courses and textbook. Note that the inner product between two vectors are defined on arbitrary dimension, but in case of cross product it is ill-defined on some dimension. Here A,B stands for vector and f,g stands for scalar. The purpose of this post is to show following vector-identities This is an English version of my former post ![]()
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