EINSTEIN Summation notation

[iontail]

hi i am just reading some notes on tesor analysis and in the notes itself while representing vectors in terms of basis using einstein summation notation the author switches between subsripts and superscripts at times. are there any different in these notation. if so what are they and when should they be used?

An examples is given on the pdf in section 1.3.

 

[Fredrik]

They're using subscripts to label different members of the same basis, and superscripts to label components of a vector, so $$A=A^i e_i$$. The components of the metric are $$g_{ij}=g(e_i,e_j)$$. The $$g^{ij}$$ are the components of the inverse of the matrix with components $$g_{ij}$$. Therefore $$g^{ij}g_{jk}=\delta^i_k$$. The $$e^i$$ are members of a basis for the dual space of the vector space with basis $$\{e_i\}$$. They're defining them by $$e^i=g^{ij}e_j$$. An expansion of a member of the dual space in terms of the $$e^i$$ would appear as $$\omega=\omega_i e^i$$, i.e. components of dual vectors are labeled by a subscript.

The dual space V* of a real vector space V is the set of continuous linear functions from V into the real numbers.

 

[iontail]

hi thanks for you reply. what is the underlying benifit of switching between the vectors on covector (indices). I ask this because Feynman introduces the concept of four vectors only using subscripts only.

 

[Fredrik]

In this context, there is no advantage at all. You can think of these equations as matrix equations, put all the indices downstairs, and forget you've ever even heard the word "tensor". (I often do that myself. See this for example. But then you should also be aware of this so that you understand what you read in books. Note in particular the expression for the inverse of a Lorentz transformation).

In differential geometry, the distinction between subscripts and superscripts has the advantage that the notation reveals what sort of tensor you're dealing with. For example, when you see $R_{abc}{}^d$, you know it's supposed to be acting on three tangent vectors and one cotangent vector.

 

[blue_raver22]

Hello there,

Einstein summation notation is a convenient way to represent vector and tensor equations in a concise and general form. In this notation, repeated indices are implicitly summed over, which simplifies the expressions and makes them easier to manipulate. The use of subscripts and superscripts in this notation is interchangeable, and the choice often depends on personal preference or the specific context of the equation.

In general, subscripts are used to indicate the position of a component in a vector or tensor, while superscripts are used to indicate the type or order of the object. For example, in the expression for a vector V, V_i would represent the ith component of the vector, while V^i would represent the contravariant component of the vector.

It is important to note that the use of subscripts and superscripts should be consistent within a particular equation or problem, in order to avoid confusion. In some cases, it may also be necessary to use both subscripts and superscripts to fully represent a tensor, such as in the case of a mixed tensor.

In summary, the use of subscripts and superscripts in Einstein summation notation is a matter of convention and personal preference, but it is important to be consistent and clear in their usage to accurately represent the equations and tensors being discussed.