How to write an integral constraint in the form of matrix linear equations?

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I have the following linear systems of equations for
$$
boldsymbol u_K = (u_K^i)_i = 1^3, ~~ u^i_K = boldsymbol c^i cdot boldsymbol varphi = sum_k = 1^K c_K^i varphi_kleft(boldsymbol xright),
$$
where $boldsymbol c^i$, $i = 1, 2, 3$, are unknown constant vectors, $boldsymbol varphi$ are given functions:
$$
int_Omega nablagammaleft(nablaboldsymbol u_Kright) varphi_k'left(boldsymbol xright) rm d boldsymbol x = 0, ~~ k' = 1, dots, K,
$$
where $Omega in mathbbR^3$,
$$
gammaleft(nablaboldsymbol u_Kright) = alpha operatornametrleft[nabla boldsymbol u_Kleft(boldsymbol xright)right] rm Id_3 times 3 + beta left[nabla boldsymbol u_Kleft(boldsymbol xright) + left(nabla boldsymbol u_Kleft(boldsymbol xright)right)^rm Tright],
$$
and $rm Id_3 times 3$ is the $3 times 3$ identity matrix, $alpha$, $beta = const$.



Since the system is linear in $boldsymbol u_K$, then the integration will lead to a linear system of algebraic equations for $boldsymbol c^i$, $i = 1, 2, 3$:
$$
bf A^i boldsymbol c^i = 0.
$$
What do the matrices $bf A^i$ look like? Because of my little knowledge of vector algebra, sums and dot products make me confused.




linear-algebra integration matrices systems-of-equations




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    up vote
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    down vote

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    I have the following linear systems of equations for
    $$
    boldsymbol u_K = (u_K^i)_i = 1^3, ~~ u^i_K = boldsymbol c^i cdot boldsymbol varphi = sum_k = 1^K c_K^i varphi_kleft(boldsymbol xright),
    $$
    where $boldsymbol c^i$, $i = 1, 2, 3$, are unknown constant vectors, $boldsymbol varphi$ are given functions:
    $$
    int_Omega nablagammaleft(nablaboldsymbol u_Kright) varphi_k'left(boldsymbol xright) rm d boldsymbol x = 0, ~~ k' = 1, dots, K,
    $$
    where $Omega in mathbbR^3$,
    $$
    gammaleft(nablaboldsymbol u_Kright) = alpha operatornametrleft[nabla boldsymbol u_Kleft(boldsymbol xright)right] rm Id_3 times 3 + beta left[nabla boldsymbol u_Kleft(boldsymbol xright) + left(nabla boldsymbol u_Kleft(boldsymbol xright)right)^rm Tright],
    $$
    and $rm Id_3 times 3$ is the $3 times 3$ identity matrix, $alpha$, $beta = const$.



    Since the system is linear in $boldsymbol u_K$, then the integration will lead to a linear system of algebraic equations for $boldsymbol c^i$, $i = 1, 2, 3$:
    $$
    bf A^i boldsymbol c^i = 0.
    $$
    What do the matrices $bf A^i$ look like? Because of my little knowledge of vector algebra, sums and dot products make me confused.




    linear-algebra integration matrices systems-of-equations




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have the following linear systems of equations for
      $$
      boldsymbol u_K = (u_K^i)_i = 1^3, ~~ u^i_K = boldsymbol c^i cdot boldsymbol varphi = sum_k = 1^K c_K^i varphi_kleft(boldsymbol xright),
      $$
      where $boldsymbol c^i$, $i = 1, 2, 3$, are unknown constant vectors, $boldsymbol varphi$ are given functions:
      $$
      int_Omega nablagammaleft(nablaboldsymbol u_Kright) varphi_k'left(boldsymbol xright) rm d boldsymbol x = 0, ~~ k' = 1, dots, K,
      $$
      where $Omega in mathbbR^3$,
      $$
      gammaleft(nablaboldsymbol u_Kright) = alpha operatornametrleft[nabla boldsymbol u_Kleft(boldsymbol xright)right] rm Id_3 times 3 + beta left[nabla boldsymbol u_Kleft(boldsymbol xright) + left(nabla boldsymbol u_Kleft(boldsymbol xright)right)^rm Tright],
      $$
      and $rm Id_3 times 3$ is the $3 times 3$ identity matrix, $alpha$, $beta = const$.



      Since the system is linear in $boldsymbol u_K$, then the integration will lead to a linear system of algebraic equations for $boldsymbol c^i$, $i = 1, 2, 3$:
      $$
      bf A^i boldsymbol c^i = 0.
      $$
      What do the matrices $bf A^i$ look like? Because of my little knowledge of vector algebra, sums and dot products make me confused.




      linear-algebra integration matrices systems-of-equations




      share|cite|improve this question











      I have the following linear systems of equations for
      $$
      boldsymbol u_K = (u_K^i)_i = 1^3, ~~ u^i_K = boldsymbol c^i cdot boldsymbol varphi = sum_k = 1^K c_K^i varphi_kleft(boldsymbol xright),
      $$
      where $boldsymbol c^i$, $i = 1, 2, 3$, are unknown constant vectors, $boldsymbol varphi$ are given functions:
      $$
      int_Omega nablagammaleft(nablaboldsymbol u_Kright) varphi_k'left(boldsymbol xright) rm d boldsymbol x = 0, ~~ k' = 1, dots, K,
      $$
      where $Omega in mathbbR^3$,
      $$
      gammaleft(nablaboldsymbol u_Kright) = alpha operatornametrleft[nabla boldsymbol u_Kleft(boldsymbol xright)right] rm Id_3 times 3 + beta left[nabla boldsymbol u_Kleft(boldsymbol xright) + left(nabla boldsymbol u_Kleft(boldsymbol xright)right)^rm Tright],
      $$
      and $rm Id_3 times 3$ is the $3 times 3$ identity matrix, $alpha$, $beta = const$.



      Since the system is linear in $boldsymbol u_K$, then the integration will lead to a linear system of algebraic equations for $boldsymbol c^i$, $i = 1, 2, 3$:
      $$
      bf A^i boldsymbol c^i = 0.
      $$
      What do the matrices $bf A^i$ look like? Because of my little knowledge of vector algebra, sums and dot products make me confused.





      linear-algebra integration matrices systems-of-equations





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      asked Jul 21 at 6:43









      Asatur Khurshudyan

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