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I am agonizing about the following integral identity:

[tex]

\frac{d}{dt} \int \int_{g(x,y) \leq t} f(x,y) dx dy = \int_{g(x,y)=t} f(x,y) \frac{1}{\left| \nabla g(x,y) \right|} ds,

[/tex]

where ds is the line element. Clearly, using the Heavisite step function, the condition [tex]g(x,y) \leq t[/tex] is transferred into the integrand. Differentiation with respect to t yields a Dirac delta-function. However, how can I eventually arrive at the line integral. The picture is quite clear, if one imagines a domain defined by [tex]g(x,y) \leq t[/tex] which is growing with t. The increase in the integral can then be evaluated by summing contributions along its circumference with [tex]\left| \nabla g(x,y) \right|[/tex] giving the density of isolines.

How can one formally obtain this result?

Thank you for you help,

Daniel

[tex]

\frac{d}{dt} \int \int_{g(x,y) \leq t} f(x,y) dx dy = \int_{g(x,y)=t} f(x,y) \frac{1}{\left| \nabla g(x,y) \right|} ds,

[/tex]

where ds is the line element. Clearly, using the Heavisite step function, the condition [tex]g(x,y) \leq t[/tex] is transferred into the integrand. Differentiation with respect to t yields a Dirac delta-function. However, how can I eventually arrive at the line integral. The picture is quite clear, if one imagines a domain defined by [tex]g(x,y) \leq t[/tex] which is growing with t. The increase in the integral can then be evaluated by summing contributions along its circumference with [tex]\left| \nabla g(x,y) \right|[/tex] giving the density of isolines.

How can one formally obtain this result?

Thank you for you help,

Daniel

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