Some examples of applying the Einstein summation notation.
Example 1. Let us consider the quantity
for a three dimensional space. Since the index
occurs as both a subscript and a superscript, we sum on
from 1 to 3. This yields
Now each term of
is such that
is both a subscript and superscript. Summing on
from 1 to 3 as prescribed by our summation convention yields the quadratic form
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Example 2. If
is a set of independent variables, then
and if
We may write
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(1) |
The symbol
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
,
,
, and
to be constant. Differentiating
with respect to a given variable, say
, yields
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Now differentiating with respect to
yields
so that
or
for
.
Example 3. We define
, to have the following numerical values: Let
. We now consider the expression
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(2) |
Expanding (2) by use of our summation convention yields
The reader who is familiar with second-order determinants quickly recognizes that
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(3) |
Example 4. The system of equations
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(4) |
represents a coordinate transformation from an
coordinate system to a
coordinate system. From the calculus we have
The
in the term
is to be considered as a subscript. If, furthermore, the
,
, can be solved for the
, and assuming differentiability of the
with respect to each
, one obtains
Differentiating this expression with respect to
yields
Multiplying both sides of this equation by
amd summing on the inex
yields
or
which yields
In particular, if
, then
[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].
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