Motivation usage of Gramian Matrix for Integration on Submanifolds












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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.










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    $begingroup$


    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.










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      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.










      share|cite|improve this question











      $endgroup$




      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






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      edited Jan 11 at 13:50









      Namaste

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      asked Jan 11 at 12:46









      SlyderSlyder

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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.






          share|cite|improve this 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.






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              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.






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                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.






                share|cite|improve this answer









                $endgroup$



                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.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Feb 1 at 7:34









                eddieeddie

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