differential of integration over fibers
$begingroup$
Is there a way to express the differential of a fiber integral that is similar to Reynolds transport problem or Leibniz rule? Here is the following setting:
Let $pi: Xmapsto Y$ be a fiber bundle on two compact connected manifolds and $f$ a smooth function on $X$.
By the co-area formula we have that:
$int_X f(x)dx = int_Y int_{F_y}f(x)NJpi(x) ^{-1}dF_y dy$
where $F_y = pi^{-1}(y)$ and $NJpi(x)$ is the normal jacobian $NJpi(x) = det (dpi_xcirc dpi_x^*)^{frac{1}{2}}$.
One can consider the fiber integral:
$g(y) = int_{F_y}f(x)NJpi(x) ^{-1}dF_y$
The question is the following: Is there an expression for the differential of $g$: $d_yg$ in terms of $f$ and $pi$?
integration derivatives differential-geometry fiber-bundles
asked Dec 29 '18 at 13:59
Michael ArbelMichael Arbel
163
$endgroup$
add a comment |
$begingroup$
Is there a way to express the differential of a fiber integral that is similar to Reynolds transport problem or Leibniz rule? Here is the following setting:
Let $pi: Xmapsto Y$ be a fiber bundle on two compact connected manifolds and $f$ a smooth function on $X$.
By the co-area formula we have that:
$int_X f(x)dx = int_Y int_{F_y}f(x)NJpi(x) ^{-1}dF_y dy$
where $F_y = pi^{-1}(y)$ and $NJpi(x)$ is the normal jacobian $NJpi(x) = det (dpi_xcirc dpi_x^*)^{frac{1}{2}}$.
One can consider the fiber integral:
$g(y) = int_{F_y}f(x)NJpi(x) ^{-1}dF_y$
The question is the following: Is there an expression for the differential of $g$: $d_yg$ in terms of $f$ and $pi$?
integration derivatives differential-geometry fiber-bundles
asked Dec 29 '18 at 13:59
Michael ArbelMichael Arbel
163
$endgroup$
add a comment |
$begingroup$
Is there a way to express the differential of a fiber integral that is similar to Reynolds transport problem or Leibniz rule? Here is the following setting:
Let $pi: Xmapsto Y$ be a fiber bundle on two compact connected manifolds and $f$ a smooth function on $X$.
By the co-area formula we have that:
$int_X f(x)dx = int_Y int_{F_y}f(x)NJpi(x) ^{-1}dF_y dy$
where $F_y = pi^{-1}(y)$ and $NJpi(x)$ is the normal jacobian $NJpi(x) = det (dpi_xcirc dpi_x^*)^{frac{1}{2}}$.
One can consider the fiber integral:
$g(y) = int_{F_y}f(x)NJpi(x) ^{-1}dF_y$
The question is the following: Is there an expression for the differential of $g$: $d_yg$ in terms of $f$ and $pi$?
integration derivatives differential-geometry fiber-bundles
asked Dec 29 '18 at 13:59
Michael ArbelMichael Arbel
163
$endgroup$
Is there a way to express the differential of a fiber integral that is similar to Reynolds transport problem or Leibniz rule? Here is the following setting:
Let $pi: Xmapsto Y$ be a fiber bundle on two compact connected manifolds and $f$ a smooth function on $X$.
By the co-area formula we have that:
$int_X f(x)dx = int_Y int_{F_y}f(x)NJpi(x) ^{-1}dF_y dy$
where $F_y = pi^{-1}(y)$ and $NJpi(x)$ is the normal jacobian $NJpi(x) = det (dpi_xcirc dpi_x^*)^{frac{1}{2}}$.
One can consider the fiber integral:
$g(y) = int_{F_y}f(x)NJpi(x) ^{-1}dF_y$
The question is the following: Is there an expression for the differential of $g$: $d_yg$ in terms of $f$ and $pi$?
integration derivatives differential-geometry fiber-bundles
integration derivatives differential-geometry fiber-bundles
asked Dec 29 '18 at 13:59
Michael ArbelMichael Arbel
163
asked Dec 29 '18 at 13:59
Michael ArbelMichael Arbel
163
edited Jan 29 at 1:19
Michael Arbel
asked Dec 29 '18 at 13:59
Michael ArbelMichael Arbel
163
asked Dec 29 '18 at 13:59
Michael ArbelMichael Arbel
163
asked Dec 29 '18 at 13:59
Michael ArbelMichael Arbel
163
163
add a comment |
add a comment |
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