Some examples of applying the Einstein summation notation.

Example 1. Let us consider the quantity

for a three dimensional space. Since the index α occurs as both a subscript and a superscript, we sum on α from 1 to 3. This yields

Now each term of S is such that β is both a subscript and superscript. Summing on β from 1 to 3 as prescribed by our summation convention yields the quadratic form

Example 2. If x¹,x²,x³,…,xⁿ is a set of independent variables, then

and if i≠j

We may write

(1)

The symbol δ_jⁱ is called the Kronecker delta. We have

Let us now assume that the quadratic form at the end of example 1 vanishes identically for all values of the independent variables x¹,x², x³, and a _ij to be constant. Differentiating S = a_αβx^αx^β = 0 with respect to a given variable, say xⁱ, yields

Now differentiating with respect to xⁱ yields

so that a_ji + a_ij = 0 or a_ij = −a_ji for i,j = 1, 2, 3.

Example 3. We define 𝜖^ij,i,j = 1, 2, to have the following numerical values: Let 𝜖¹¹ = 𝜖²² = 0,𝜖¹² = 1,𝜖²¹ = −1. We now consider the expression

(2)

Expanding (2) by use of our summation convention yields

The reader who is familiar with second-order determinants quickly recognizes that

(3)

Example 4. The system of equations

(4)

represents a coordinate transformation from an (x¹,x²,,xⁿ) coordinate system to a (y¹,y²,,yⁿ) coordinate system. From the calculus we have

The α in the term is to be considered as a subscript. If, furthermore, the xⁱ, i = 1, 2,,n, can be solved for the y¹,y²,,yⁿ, and assuming differentiability of the xⁱ with respect to each yⁱ, one obtains

Differentiating this expression with respect to y^k yields

Multiplying both sides of this equation by amd summing on the inex i yields

or

which yields

In particular, if y = f(x), then

0.1 References

[1] Lass, Harry. ”Elements of pure and applied mathematics” New York: McGraw-Hill Companies, 1957.

This entry is a derivative of the Public domain work [1].