Brute Force Vector Field Transformations

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I want to know if there is a way to see vector transformations in a more foolproof, brute force way. For example, I know that if we rotate a vector field with a rotation matrix $R$, we transform as follows:



$$vecV(vecx) to RvecV(R^-1vecx)$$



I intuitively see why, but is there an explicit way to show this? Does it not make sense to think about the matrix acting on the vector field, since each point $vecx$ defines a new vector space? I was thinking of just a Taylor expansion:



$$vecV(vecx)=vecV(0)+vecxcdotnablavecV(0)+...$$



And perhaps a factor of $R$ acts on the whole thing, but it's not clear to me if that works...




multivariable-calculus derivatives vector-spaces vectors vector-bundles




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    I want to know if there is a way to see vector transformations in a more foolproof, brute force way. For example, I know that if we rotate a vector field with a rotation matrix $R$, we transform as follows:



    $$vecV(vecx) to RvecV(R^-1vecx)$$



    I intuitively see why, but is there an explicit way to show this? Does it not make sense to think about the matrix acting on the vector field, since each point $vecx$ defines a new vector space? I was thinking of just a Taylor expansion:



    $$vecV(vecx)=vecV(0)+vecxcdotnablavecV(0)+...$$



    And perhaps a factor of $R$ acts on the whole thing, but it's not clear to me if that works...




    multivariable-calculus derivatives vector-spaces vectors vector-bundles




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I want to know if there is a way to see vector transformations in a more foolproof, brute force way. For example, I know that if we rotate a vector field with a rotation matrix $R$, we transform as follows:



      $$vecV(vecx) to RvecV(R^-1vecx)$$



      I intuitively see why, but is there an explicit way to show this? Does it not make sense to think about the matrix acting on the vector field, since each point $vecx$ defines a new vector space? I was thinking of just a Taylor expansion:



      $$vecV(vecx)=vecV(0)+vecxcdotnablavecV(0)+...$$



      And perhaps a factor of $R$ acts on the whole thing, but it's not clear to me if that works...




      multivariable-calculus derivatives vector-spaces vectors vector-bundles




      share|cite|improve this question











      I want to know if there is a way to see vector transformations in a more foolproof, brute force way. For example, I know that if we rotate a vector field with a rotation matrix $R$, we transform as follows:



      $$vecV(vecx) to RvecV(R^-1vecx)$$



      I intuitively see why, but is there an explicit way to show this? Does it not make sense to think about the matrix acting on the vector field, since each point $vecx$ defines a new vector space? I was thinking of just a Taylor expansion:



      $$vecV(vecx)=vecV(0)+vecxcdotnablavecV(0)+...$$



      And perhaps a factor of $R$ acts on the whole thing, but it's not clear to me if that works...





      multivariable-calculus derivatives vector-spaces vectors vector-bundles





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 24 at 23:48









      Connor Dolan

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