Einstein summation convention

Einstein summation convention

The

Einstein summation convention

implies that when an index occurs
more than once in the same expression, the expression is implicitly
summed over all possible values for that index. Therefore, in order to
use the

summation

convention, it must be clear from the context
over what range indices should be summed.

The Einstein summation convention is illustrated in the below examples.

1.

Let

{

e

i

}

i

=

1

n

be a

orthogonal basis

in

ℝ

n

.
Then the

inner product

of the vectors

u

=

u

i

⁢

e

i

=

(

∑

i

=

1

n

)

⁡

u

i

⁢

e

i

and

v

=

v

i

⁢

e

i

=

(

∑

i

=

1

n

)

⁡

v

i

⁢

e

i

, is

u

⋅

v

=

u

i

⁢

v

j

⁢

e

i

⋅

e

j

=

δ

i

⁢

j

⁢

u

i

⁢

v

j

.

2.

Let

V

be a

vector space

with basis

{

e

i

}

i

=

1

n

and
a

dual basis

{

e

i

}

i

=

1

n

. Then, for a vector

v

=

v

i

⁢

e

i

and dual vectors

α

=

α

i

⁢

e

i

and

β

=

β

i

⁢

e

i

, we have

(

α

+

β

)

⁢

(

v

)

=

α

i

⁢

v

i

+

β

j

⁢

v

j

=

(

α

i

+

β

i

)

⁢

v

i

.

This example shows that the summation convention is “distributive” in
a natural way.

3.

Chain rule

. Let

F

:

ℝ

m

→

ℝ

n

,

x

↦

(

F

1

,

⋯

,

F

n

)

, and

G

:

ℝ

n

→

ℝ

p

,

y

↦

(

G

1

⁢

(

y

)

,

⋯

,

G

p

⁢

(

y

)

)

be

smooth functions

. Then

∂

(

G

∘

F

)

i

∂

⁡

x

j

⁢

(

x

)

=

∂

⁡

G

i

∂

⁡

y

k

⁢

(

F

⁢

(

x

)

)

⁢

∂

⁡

F

k

∂

⁡

x

j

⁢

(

x

)

,

where the right hand side is summed over

k

=

1

,

…

,

n

.

An index which is summed is called a

dummy index

or

dummy variable

.
For instance,

i

is a dummy index in

v

i

⁢

e

i

. An expression does not depend on
a dummy index, i.e.,

v

i

⁢

e

i

=

v

j

⁢

e

j

.
It is common that one must change the name of dummy indices. For
instance, above, in Example 2 when we calculated

u

⋅

v

, it was necessary to change the index

i

in

v

=

v

i

⁢

e

i

to

j

so that it would not clash with

u

=

u

i

⁢

e

i

.

When using the Einstein summation convention, objects are usually indexed
so that when summing, one index will always be an “upper index” and
the other a “lower index”.
Then summing should only take place over upper and lower indices.
In the above examples, we have followed
this rule. Therefore we did not write

δ

i

⁢

j

⁢

u

i

⁢

v

j

=

u

i

⁢

v

i

in the first example since

u

i

⁢

v

i

has two upper indices. This is

consistent

; it is not possible
to take the inner product of two vectors without a metric, which is here

δ

i

⁢

j

.
The last example illustrates that when we consider

k

as a “lower
index” in

∂

⁡

G

i

∂

⁡

y

k

, then the chain rule
obeys this upper-lower rule for the indices.

Title

Einstein summation convention

Canonical name

EinsteinSummationConvention

Date of creation

2013-03-22 13:32:03

Last modified on

2013-03-22 13:32:03

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 15A69

Related topic

Summation

Defines

dummy index

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