Mixing
Integrating the composition of a Heaviside function with a smooth function
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I am trying to find how to compute an integral of the form:
$int_{R^n}{Theta(g(x))f(x),dx}$, where $Theta$ is the Heaviside function, $g(x)$ is a smooth function (a result for more general $g$ is also welcome) and you can assume that $f(x)$ is also smooth. For the Dirac delta (which is the derivative of $Theta$) we have the relationship:
$$int_{R^n}{f(x)delta(g(x))|nabla g(x)|,dx} = int_{R^n}{f(x)delta_S(x),dx} = int_{S}{f(x),dsigma(x)}$$
Where $S = {x|g(x) = 0, x in R^n}$, and $sigma(x)$ is the surface measure on $S$. Is there a similar relationship for the Heaviside function? Or any standard method that would help me compute integrals of that form. References on the subject are welcome, but keep in mind I am a computer science student (so my mathematical background is fairly limited).
multivariable-calculus dirac-delta geometric-measure-theory step-function
asked Jan 25 at 19:01
lightxbulblightxbulb
1,140311
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add a comment |
$begingroup$
I am trying to find how to compute an integral of the form:
$int_{R^n}{Theta(g(x))f(x),dx}$, where $Theta$ is the Heaviside function, $g(x)$ is a smooth function (a result for more general $g$ is also welcome) and you can assume that $f(x)$ is also smooth. For the Dirac delta (which is the derivative of $Theta$) we have the relationship:
$$int_{R^n}{f(x)delta(g(x))|nabla g(x)|,dx} = int_{R^n}{f(x)delta_S(x),dx} = int_{S}{f(x),dsigma(x)}$$
Where $S = {x|g(x) = 0, x in R^n}$, and $sigma(x)$ is the surface measure on $S$. Is there a similar relationship for the Heaviside function? Or any standard method that would help me compute integrals of that form. References on the subject are welcome, but keep in mind I am a computer science student (so my mathematical background is fairly limited).
multivariable-calculus dirac-delta geometric-measure-theory step-function
asked Jan 25 at 19:01
lightxbulblightxbulb
1,140311
$endgroup$
add a comment |
$begingroup$
I am trying to find how to compute an integral of the form:
$int_{R^n}{Theta(g(x))f(x),dx}$, where $Theta$ is the Heaviside function, $g(x)$ is a smooth function (a result for more general $g$ is also welcome) and you can assume that $f(x)$ is also smooth. For the Dirac delta (which is the derivative of $Theta$) we have the relationship:
$$int_{R^n}{f(x)delta(g(x))|nabla g(x)|,dx} = int_{R^n}{f(x)delta_S(x),dx} = int_{S}{f(x),dsigma(x)}$$
Where $S = {x|g(x) = 0, x in R^n}$, and $sigma(x)$ is the surface measure on $S$. Is there a similar relationship for the Heaviside function? Or any standard method that would help me compute integrals of that form. References on the subject are welcome, but keep in mind I am a computer science student (so my mathematical background is fairly limited).
multivariable-calculus dirac-delta geometric-measure-theory step-function
asked Jan 25 at 19:01
lightxbulblightxbulb
1,140311
$endgroup$
I am trying to find how to compute an integral of the form:
$int_{R^n}{Theta(g(x))f(x),dx}$, where $Theta$ is the Heaviside function, $g(x)$ is a smooth function (a result for more general $g$ is also welcome) and you can assume that $f(x)$ is also smooth. For the Dirac delta (which is the derivative of $Theta$) we have the relationship:
$$int_{R^n}{f(x)delta(g(x))|nabla g(x)|,dx} = int_{R^n}{f(x)delta_S(x),dx} = int_{S}{f(x),dsigma(x)}$$
Where $S = {x|g(x) = 0, x in R^n}$, and $sigma(x)$ is the surface measure on $S$. Is there a similar relationship for the Heaviside function? Or any standard method that would help me compute integrals of that form. References on the subject are welcome, but keep in mind I am a computer science student (so my mathematical background is fairly limited).
multivariable-calculus dirac-delta geometric-measure-theory step-function
multivariable-calculus dirac-delta geometric-measure-theory step-function
asked Jan 25 at 19:01
lightxbulblightxbulb
1,140311
asked Jan 25 at 19:01
lightxbulblightxbulb
1,140311
asked Jan 25 at 19:01
lightxbulblightxbulb
1,140311
asked Jan 25 at 19:01
lightxbulblightxbulb
1,140311
asked Jan 25 at 19:01
lightxbulblightxbulb
1,140311
1,140311
add a comment |
add a comment |
1 Answer
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$begingroup$
Turns out there is no such relationship for the Heaviside step function (and it's not necessary). We can directly apply it to the integration domain. That is:
$$int_{R^n}{Theta(g(x))f(x),dx} = int_{{x|g(x)geq 0}}{f(x),dx}$$
answered Feb 8 at 18:24
lightxbulblightxbulb
1,140311
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Turns out there is no such relationship for the Heaviside step function (and it's not necessary). We can directly apply it to the integration domain. That is:
$$int_{R^n}{Theta(g(x))f(x),dx} = int_{{x|g(x)geq 0}}{f(x),dx}$$
answered Feb 8 at 18:24
lightxbulblightxbulb
1,140311
$endgroup$
add a comment |
$begingroup$
Turns out there is no such relationship for the Heaviside step function (and it's not necessary). We can directly apply it to the integration domain. That is:
$$int_{R^n}{Theta(g(x))f(x),dx} = int_{{x|g(x)geq 0}}{f(x),dx}$$
answered Feb 8 at 18:24
lightxbulblightxbulb
1,140311
$endgroup$
add a comment |
$begingroup$
Turns out there is no such relationship for the Heaviside step function (and it's not necessary). We can directly apply it to the integration domain. That is:
$$int_{R^n}{Theta(g(x))f(x),dx} = int_{{x|g(x)geq 0}}{f(x),dx}$$
answered Feb 8 at 18:24
lightxbulblightxbulb
1,140311
$endgroup$
Turns out there is no such relationship for the Heaviside step function (and it's not necessary). We can directly apply it to the integration domain. That is:
$$int_{R^n}{Theta(g(x))f(x),dx} = int_{{x|g(x)geq 0}}{f(x),dx}$$
answered Feb 8 at 18:24
lightxbulblightxbulb
1,140311
answered Feb 8 at 18:24
lightxbulblightxbulb
1,140311
answered Feb 8 at 18:24
lightxbulblightxbulb
1,140311
answered Feb 8 at 18:24
lightxbulblightxbulb
1,140311
1,140311
add a comment |
add a comment |
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