Motivation usage of Gramian Matrix for Integration on Submanifolds
$begingroup$
I am struggling to understand the motivation for the definition of the Gram Matrix and its corresponding role for the definition of the Lebesgue measure on submanifolds.
$M subset mathbb{R}^n k$ dimensional $C^1$-Submanifold
$phi^{-1}: T subseteq mathbb{R}^m to M$ local parametrization of M
We define the Gram-Matrix as
$G_phi: T to mathbb{R}^{ntimes n}$
$t mapsto (D(phi^{-1}(t))^TD(phi^{-1}(t)) $
and its corresponding determinant at $tin T: g_phi(t)$.
Using this one defines the m-dimensional-Lebesgue measure $lambda^m$ on the Submanifold M (for simplification assume that there exists a map which already describes all of $M$, i.e.:
$ phi: M mapsto T subset mathbb{R}^m $, the Atlas of M is only one map.)
Then one defines:
$lambda_M: mathbb{B}^d cap M to [0,infty]$
$ B mapsto int_{phi(B)} (g_{ phi^{-1}}(t))^{1/2}dlambda^m(t) $
I think the usage of the Gram-Determinant accounts for some deformation of $phi$, similar to the Transformation Rule, but the construction of the Gram-Matrix does not make sense to me yet.
integration measure-theory lebesgue-measure vector-analysis submanifold
edited Jan 11 at 13:50
Namaste
1
asked Jan 11 at 12:46
SlyderSlyder
316
$endgroup$
add a comment |
$begingroup$
I am struggling to understand the motivation for the definition of the Gram Matrix and its corresponding role for the definition of the Lebesgue measure on submanifolds.
$M subset mathbb{R}^n k$ dimensional $C^1$-Submanifold
$phi^{-1}: T subseteq mathbb{R}^m to M$ local parametrization of M
We define the Gram-Matrix as
$G_phi: T to mathbb{R}^{ntimes n}$
$t mapsto (D(phi^{-1}(t))^TD(phi^{-1}(t)) $
and its corresponding determinant at $tin T: g_phi(t)$.
Using this one defines the m-dimensional-Lebesgue measure $lambda^m$ on the Submanifold M (for simplification assume that there exists a map which already describes all of $M$, i.e.:
$ phi: M mapsto T subset mathbb{R}^m $, the Atlas of M is only one map.)
Then one defines:
$lambda_M: mathbb{B}^d cap M to [0,infty]$
$ B mapsto int_{phi(B)} (g_{ phi^{-1}}(t))^{1/2}dlambda^m(t) $
I think the usage of the Gram-Determinant accounts for some deformation of $phi$, similar to the Transformation Rule, but the construction of the Gram-Matrix does not make sense to me yet.
integration measure-theory lebesgue-measure vector-analysis submanifold
edited Jan 11 at 13:50
Namaste
1
asked Jan 11 at 12:46
SlyderSlyder
316
$endgroup$
add a comment |
$begingroup$
I am struggling to understand the motivation for the definition of the Gram Matrix and its corresponding role for the definition of the Lebesgue measure on submanifolds.
$M subset mathbb{R}^n k$ dimensional $C^1$-Submanifold
$phi^{-1}: T subseteq mathbb{R}^m to M$ local parametrization of M
We define the Gram-Matrix as
$G_phi: T to mathbb{R}^{ntimes n}$
$t mapsto (D(phi^{-1}(t))^TD(phi^{-1}(t)) $
and its corresponding determinant at $tin T: g_phi(t)$.
Using this one defines the m-dimensional-Lebesgue measure $lambda^m$ on the Submanifold M (for simplification assume that there exists a map which already describes all of $M$, i.e.:
$ phi: M mapsto T subset mathbb{R}^m $, the Atlas of M is only one map.)
Then one defines:
$lambda_M: mathbb{B}^d cap M to [0,infty]$
$ B mapsto int_{phi(B)} (g_{ phi^{-1}}(t))^{1/2}dlambda^m(t) $
I think the usage of the Gram-Determinant accounts for some deformation of $phi$, similar to the Transformation Rule, but the construction of the Gram-Matrix does not make sense to me yet.
integration measure-theory lebesgue-measure vector-analysis submanifold
edited Jan 11 at 13:50
Namaste
1
asked Jan 11 at 12:46
SlyderSlyder
316
$endgroup$
I am struggling to understand the motivation for the definition of the Gram Matrix and its corresponding role for the definition of the Lebesgue measure on submanifolds.
$M subset mathbb{R}^n k$ dimensional $C^1$-Submanifold
$phi^{-1}: T subseteq mathbb{R}^m to M$ local parametrization of M
We define the Gram-Matrix as
$G_phi: T to mathbb{R}^{ntimes n}$
$t mapsto (D(phi^{-1}(t))^TD(phi^{-1}(t)) $
and its corresponding determinant at $tin T: g_phi(t)$.
Using this one defines the m-dimensional-Lebesgue measure $lambda^m$ on the Submanifold M (for simplification assume that there exists a map which already describes all of $M$, i.e.:
$ phi: M mapsto T subset mathbb{R}^m $, the Atlas of M is only one map.)
Then one defines:
$lambda_M: mathbb{B}^d cap M to [0,infty]$
$ B mapsto int_{phi(B)} (g_{ phi^{-1}}(t))^{1/2}dlambda^m(t) $
I think the usage of the Gram-Determinant accounts for some deformation of $phi$, similar to the Transformation Rule, but the construction of the Gram-Matrix does not make sense to me yet.
integration measure-theory lebesgue-measure vector-analysis submanifold
integration measure-theory lebesgue-measure vector-analysis submanifold
edited Jan 11 at 13:50
Namaste
1
asked Jan 11 at 12:46
SlyderSlyder
316
edited Jan 11 at 13:50
Namaste
1
asked Jan 11 at 12:46
SlyderSlyder
316
edited Jan 11 at 13:50
Namaste
1
edited Jan 11 at 13:50
Namaste
1
edited Jan 11 at 13:50
Namaste
1
1
asked Jan 11 at 12:46
SlyderSlyder
316
asked Jan 11 at 12:46
SlyderSlyder
316
asked Jan 11 at 12:46
SlyderSlyder
316
316
add a comment |
add a comment |
1 Answer
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$begingroup$
Recall that for linear independent vectors $v_1, cdots, v_k subset mathbb{R}^n$ the $k$-dimensional volume of the k-parallelepiped $P$ is given by $Vol_{k} = sqrt{det{A^TA}}$ where $A subset mathbb{R}^{n times k}$ is the matrix containing the column vectors $v_1, cdots, v_k$ (we assume $k leq n$).
Let $M subset mathbb{R}^n$ be a $k$-dimensional submanifold with one atlas $Phi: V to M$. Let's assume we devide $V$ into finitely many small cubes $Q_i$. We might want to write $Vol_k(M) approx sum_i Vol_k(phi(Q_i))$ similarly to Riemann-sums. If we make the cubes "small enough" we have $phi approx Dphi$ since the Jacobi Matrix is the liniarization of $phi$.
Now replacing the sum by an integral and $Vol_k(..)$ by $sqrt{Dphi^T D phi}$ yields the representation. I hope this gives some motivation on the usage of the gram matrix.
answered Feb 1 at 7:34
eddieeddie
525110
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
Recall that for linear independent vectors $v_1, cdots, v_k subset mathbb{R}^n$ the $k$-dimensional volume of the k-parallelepiped $P$ is given by $Vol_{k} = sqrt{det{A^TA}}$ where $A subset mathbb{R}^{n times k}$ is the matrix containing the column vectors $v_1, cdots, v_k$ (we assume $k leq n$).
Let $M subset mathbb{R}^n$ be a $k$-dimensional submanifold with one atlas $Phi: V to M$. Let's assume we devide $V$ into finitely many small cubes $Q_i$. We might want to write $Vol_k(M) approx sum_i Vol_k(phi(Q_i))$ similarly to Riemann-sums. If we make the cubes "small enough" we have $phi approx Dphi$ since the Jacobi Matrix is the liniarization of $phi$.
Now replacing the sum by an integral and $Vol_k(..)$ by $sqrt{Dphi^T D phi}$ yields the representation. I hope this gives some motivation on the usage of the gram matrix.
answered Feb 1 at 7:34
eddieeddie
525110
$endgroup$
add a comment |
$begingroup$
Recall that for linear independent vectors $v_1, cdots, v_k subset mathbb{R}^n$ the $k$-dimensional volume of the k-parallelepiped $P$ is given by $Vol_{k} = sqrt{det{A^TA}}$ where $A subset mathbb{R}^{n times k}$ is the matrix containing the column vectors $v_1, cdots, v_k$ (we assume $k leq n$).
Let $M subset mathbb{R}^n$ be a $k$-dimensional submanifold with one atlas $Phi: V to M$. Let's assume we devide $V$ into finitely many small cubes $Q_i$. We might want to write $Vol_k(M) approx sum_i Vol_k(phi(Q_i))$ similarly to Riemann-sums. If we make the cubes "small enough" we have $phi approx Dphi$ since the Jacobi Matrix is the liniarization of $phi$.
Now replacing the sum by an integral and $Vol_k(..)$ by $sqrt{Dphi^T D phi}$ yields the representation. I hope this gives some motivation on the usage of the gram matrix.
answered Feb 1 at 7:34
eddieeddie
525110
$endgroup$
add a comment |
$begingroup$
Recall that for linear independent vectors $v_1, cdots, v_k subset mathbb{R}^n$ the $k$-dimensional volume of the k-parallelepiped $P$ is given by $Vol_{k} = sqrt{det{A^TA}}$ where $A subset mathbb{R}^{n times k}$ is the matrix containing the column vectors $v_1, cdots, v_k$ (we assume $k leq n$).
Let $M subset mathbb{R}^n$ be a $k$-dimensional submanifold with one atlas $Phi: V to M$. Let's assume we devide $V$ into finitely many small cubes $Q_i$. We might want to write $Vol_k(M) approx sum_i Vol_k(phi(Q_i))$ similarly to Riemann-sums. If we make the cubes "small enough" we have $phi approx Dphi$ since the Jacobi Matrix is the liniarization of $phi$.
Now replacing the sum by an integral and $Vol_k(..)$ by $sqrt{Dphi^T D phi}$ yields the representation. I hope this gives some motivation on the usage of the gram matrix.
answered Feb 1 at 7:34
eddieeddie
525110
$endgroup$
Recall that for linear independent vectors $v_1, cdots, v_k subset mathbb{R}^n$ the $k$-dimensional volume of the k-parallelepiped $P$ is given by $Vol_{k} = sqrt{det{A^TA}}$ where $A subset mathbb{R}^{n times k}$ is the matrix containing the column vectors $v_1, cdots, v_k$ (we assume $k leq n$).
Let $M subset mathbb{R}^n$ be a $k$-dimensional submanifold with one atlas $Phi: V to M$. Let's assume we devide $V$ into finitely many small cubes $Q_i$. We might want to write $Vol_k(M) approx sum_i Vol_k(phi(Q_i))$ similarly to Riemann-sums. If we make the cubes "small enough" we have $phi approx Dphi$ since the Jacobi Matrix is the liniarization of $phi$.
Now replacing the sum by an integral and $Vol_k(..)$ by $sqrt{Dphi^T D phi}$ yields the representation. I hope this gives some motivation on the usage of the gram matrix.
answered Feb 1 at 7:34
eddieeddie
525110
answered Feb 1 at 7:34
eddieeddie
525110
answered Feb 1 at 7:34
eddieeddie
525110
answered Feb 1 at 7:34
eddieeddie
525110
525110
add a comment |
add a comment |
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