# Dirac-Delta Function Property

1. Nov 7, 2013

### unscientific

I'm not sure how they got the RHS of equation 349:

where did the |y'(xj)| in the denominator come from?

According to (343) the RHS is only f(x0) which in this case is the jth term of the sum that gives y(xj) = 0..

2. Nov 7, 2013

### Simon Bridge

... um... from evaluating the integral?

I don't think that is what (343) says.

The integral you are evaluating is centered in a small range about $y_j=y(x_j)$.
(343) is for the entire y axis, and is centered about the origin.

3. Nov 10, 2013

### unscientific

I'm sorry I still don't quite understand how the |y'(xj)| in the denominator came about. How do you evaluate the integral?

4. Nov 10, 2013

### dauto

It comes from the Jacobian due to a change of variables done in order to perform the integral

5. Nov 10, 2013

### unscientific

I still don't quite understand why is there a denominator:

δ(y(x)) is zero everywhere except at the particular xj where δ(y(x)) = δ(y(xj)) = δ(0) = 1

Then the integral can be simplified to give:

∫ f(x) dx from xj-ε to xj+ε where ε is small enough such that it does not coincide with other solutions of y(xi) = 0 for some xi.

Then that should give f(xj)∫ dx from xj-ε to xj

= f(xj) * 1 (∫ dx from xj-ε to xj+ε = 1)
= f(xj)

where did the |f'(xj)| in the denominator come from?

6. Nov 10, 2013

### Simon Bridge

Isn't it zero everywhere except where $y=y_j:y_j=y(x_j)$
(What is y(x)?)

That means the $\delta(y)$ in the integrand turns into $\delta(y-y_j)$ to stay consistent.
Maths is a language - what is the math here supposed to be describing?

Now you can apply the rule.

Also take note: for a pure math interpretation...
... how did they change variables?

7. Nov 11, 2013

### dauto

That's all wrong. δ(0) isn't equal to unit. The integral is equal to unit. δ(0) itself is an undefined divergent quantity - infinite.

8. Nov 11, 2013

### dauto

They changed from an integral over dx to an integral over dy. There is a Jacobean factor.