# Delta function & kronecker delta

Can anyone tell me the difference between the Delta function and the Kronecker delta?

It seems that both are 1 at a certain point and 0 otherwise...

The delta function is a eigenfunction of x and the Kronecker delta is ...

i'm kind of confused..

## Answers and Replies

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mathman
Science Advisor
Delta function: Integral of f(x) F(x-a), where F is the delta function,
=f(a) when a is in the interval, and integral =0 if a is not in the interval.

Kronecker delta G(n-k) (usually for integer argument, not real) G=1 for n=k, =0 for n not=k.

quasar987
Science Advisor
Homework Helper
Gold Member
From what I understand, the kronecker delta is just an abreviation. For exemple, if we have a set of orthonormal vectors $\hat{e}_{i}$

Then the dot product of any two of these vectors can be expressed as

$$(\hat{e}_{n}|\hat{e}_{k}) = \left\{\begin{array}{rcl}1 \ \mbox{if} \ n=k\\ 0 \ \mbox{otherwise}\end{array}$$

So we write

$$(\hat{e}_{n}|\hat{e}_{k}) = \delta_{nk}$$

to compactly express this fact.

delta function

The delta function, delta(x), is infinite at x=0, zero everywhere else. It is what a normalized Gaussian "hump" looks like in the limit as its width goes to zero.

In contrast, Kronecker delta is not really a function at all ... more like an element of a matrix (the identity matrix). So Kronecker[ij] = 1 (if i==j), or 0 (if i!=j).

dextercioby
Science Advisor
Homework Helper
Delta-Dirac is a distribution,while Delta-Kronecker is an invariant totally symmetrical tensor of arbitrary rank.

Daniel.

aav
Also, both are used to express orthogonality, given a set of vectors, the arguments being the indices of the two vectors in question: the kronecker delta if that set is countable, the delta function if otherwise.

churi55 said:
Can anyone tell me the difference between the Delta function and the Kronecker delta?

It seems that both are 1 at a certain point and 0 otherwise...

The delta function is a eigenfunction of x and the Kronecker delta is ...

i'm kind of confused..
in an easy language, they are inherently the same (they have the same/analoguous meaning) but the Kronecker delta is the DISCRETE variant of the delta dirac distribution/functional. So the indices are discrete where they are continuous (they vary continuously) in case of the delta dirac distribution.

regards

marlon

dextercioby
Science Advisor
Homework Helper
In de Witt notation,where the D-dim delta-Dirac is supressed,one only finds the Kronecker one.However,the Dirac one is commonly understood.

Daniel.

$$\delta$$ij constitutes the identity matrix when:

($$\delta$$ij)i,jn [that's NOT j to the n]