Delta function & kronecker delta

In summary, the Delta function and Kronecker delta are both mathematical functions that are 1 at a certain point and 0 otherwise. The difference is that the Delta function is a continuous function while the Kronecker delta is a discrete function. The Delta function is an eigenfunction of x and the Kronecker delta is an invariant totally symmetrical tensor. They are both used to express orthogonality, with the Kronecker delta for countable sets and the Delta function for non-countable sets. In de Witt notation, the Kronecker delta is commonly used instead of the Delta function.
  • #1
churi55
4
0
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..
 
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  • #2
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.
 
  • #3
From what I understand, the kronecker delta is just an abreviation. For exemple, if we have a set of orthonormal vectors [itex]\hat{e}_{i}[/itex]

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

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

So we write

[tex](\hat{e}_{n}|\hat{e}_{k}) = \delta_{nk}[/tex]

to compactly express this fact.
 
  • #4
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).
 
  • #5
Delta-Dirac is a distribution,while Delta-Kronecker is an invariant totally symmetrical tensor of arbitrary rank.

Daniel.
 
  • #6
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.
 
  • #7
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
 
  • #8
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.
 
  • #9


[tex]\delta[/tex]ij constitutes the identity matrix when:

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

1. What is the difference between the Kronecker delta and the Dirac delta function?

The Kronecker delta, denoted as δij, is a discrete function that takes the value of 1 when the two indices i and j are equal, and 0 when they are not equal. It is commonly used in mathematical and physics equations to represent a specific location or state. On the other hand, the Dirac delta function, denoted as δ(x), is a continuous mathematical function that is defined to be 0 everywhere except at x = 0, where it is infinite. It is often used in calculus to represent an infinitesimal point mass or impulse.

2. How is the Kronecker delta used in linear algebra?

In linear algebra, the Kronecker delta is used as a convenient notation for the identity matrix. This is because it has a similar property as the identity matrix where it takes the value of 1 when the two indices are equal and 0 when they are not equal. It is also used to define the Kronecker product, which is a way to combine two matrices to form a larger matrix.

3. What is the physical interpretation of the Dirac delta function?

The Dirac delta function is often used in physics to represent a point source of energy or matter. This means that at a specific location, there is an infinite amount of energy or mass concentrated at a single point. This can be useful in modeling physical phenomena such as point charges in electrostatics or point masses in classical mechanics.

4. Can the Dirac delta function be integrated?

Yes, the Dirac delta function can be integrated, but it requires a special type of integration called the generalized integral or the Stieltjes integral. This type of integration takes into account the discontinuities of the function and allows for the integration of functions that are not defined in the traditional sense. It is often used to solve differential equations involving the Dirac delta function.

5. How do you graph the Dirac delta function?

Since the Dirac delta function is infinite at x = 0 and 0 everywhere else, it cannot be graphed in the traditional sense. However, it can be represented as a spike or impulse at x = 0 on a graph with the y-axis representing the value of the function and the x-axis representing the location. This is often referred to as the "delta spike" or the "delta function spike" graph.

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