How did the variable change in the Dirac-Delta function property equation?

[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..

 

[Simon Bridge]

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

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

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

j

) = 0...

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.

 

[unscientific]

... 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.

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

 

[dauto]

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

 

[unscientific]

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

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

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

Then the integral can be simplified to give:

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

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

= f(x_j) * 1 (∫ dx from x_j-ε to x_j+ε = 1)
= f(x_j)where did the |f'(x_j)| in the denominator come from?

 

[Simon Bridge]

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

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...

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

... how did they change variables?

 

[dauto]

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

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

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.

 

[dauto]

... how did they change variables?

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