Why should the position vector be noted as $Rhat{R}$ in spherical polar coordinates?












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Why should the position vector be noted as $Rhat{R}$ in spherical polar coordinates? Now i did the calculation like this: $vec R = R sintheta cosphi hat{i} + R sintheta sinphi hat{j} + R costheta hat{k}$ so now i am manipulating the unit vectors. As :- $$hat{R}= frac{frac{partial vec{R}}{partial R}}{left|frac{partial vec{R}}{partial R}right|}=sintheta cosphi hat{i} + sintheta sinphi hat{j} + costheta hat{k}$$ by doing similiar calculations i found $hat{theta}=costheta cosphi hat{i} + costheta sinphi hat{j} -sinthetahat{k}$. Similarly I found $hat{phi}= cosphi hat{i} + sinphihat{j}$ now position vector can be written as $vec R= [vec R. hat{R}]hat{theta} + [vec R. hat{theta}]hat{theta} + [vec{R},hat{phi}] hat{phi}$. Which gives me $vec{R} = Rhat{R} + Rsintheta hat{phi}$ not $Rhat{R}$ now where i am misunderstanding or miscalculating ?






vectors vector-analysis spherical-coordinates







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    $begingroup$


    Why should the position vector be noted as $Rhat{R}$ in spherical polar coordinates? Now i did the calculation like this: $vec R = R sintheta cosphi hat{i} + R sintheta sinphi hat{j} + R costheta hat{k}$ so now i am manipulating the unit vectors. As :- $$hat{R}= frac{frac{partial vec{R}}{partial R}}{left|frac{partial vec{R}}{partial R}right|}=sintheta cosphi hat{i} + sintheta sinphi hat{j} + costheta hat{k}$$ by doing similiar calculations i found $hat{theta}=costheta cosphi hat{i} + costheta sinphi hat{j} -sinthetahat{k}$. Similarly I found $hat{phi}= cosphi hat{i} + sinphihat{j}$ now position vector can be written as $vec R= [vec R. hat{R}]hat{theta} + [vec R. hat{theta}]hat{theta} + [vec{R},hat{phi}] hat{phi}$. Which gives me $vec{R} = Rhat{R} + Rsintheta hat{phi}$ not $Rhat{R}$ now where i am misunderstanding or miscalculating ?






    vectors vector-analysis spherical-coordinates







    share|cite|improve this question











    $endgroup$















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      1








      1





      $begingroup$


      Why should the position vector be noted as $Rhat{R}$ in spherical polar coordinates? Now i did the calculation like this: $vec R = R sintheta cosphi hat{i} + R sintheta sinphi hat{j} + R costheta hat{k}$ so now i am manipulating the unit vectors. As :- $$hat{R}= frac{frac{partial vec{R}}{partial R}}{left|frac{partial vec{R}}{partial R}right|}=sintheta cosphi hat{i} + sintheta sinphi hat{j} + costheta hat{k}$$ by doing similiar calculations i found $hat{theta}=costheta cosphi hat{i} + costheta sinphi hat{j} -sinthetahat{k}$. Similarly I found $hat{phi}= cosphi hat{i} + sinphihat{j}$ now position vector can be written as $vec R= [vec R. hat{R}]hat{theta} + [vec R. hat{theta}]hat{theta} + [vec{R},hat{phi}] hat{phi}$. Which gives me $vec{R} = Rhat{R} + Rsintheta hat{phi}$ not $Rhat{R}$ now where i am misunderstanding or miscalculating ?






      vectors vector-analysis spherical-coordinates







      share|cite|improve this question











      $endgroup$




      Why should the position vector be noted as $Rhat{R}$ in spherical polar coordinates? Now i did the calculation like this: $vec R = R sintheta cosphi hat{i} + R sintheta sinphi hat{j} + R costheta hat{k}$ so now i am manipulating the unit vectors. As :- $$hat{R}= frac{frac{partial vec{R}}{partial R}}{left|frac{partial vec{R}}{partial R}right|}=sintheta cosphi hat{i} + sintheta sinphi hat{j} + costheta hat{k}$$ by doing similiar calculations i found $hat{theta}=costheta cosphi hat{i} + costheta sinphi hat{j} -sinthetahat{k}$. Similarly I found $hat{phi}= cosphi hat{i} + sinphihat{j}$ now position vector can be written as $vec R= [vec R. hat{R}]hat{theta} + [vec R. hat{theta}]hat{theta} + [vec{R},hat{phi}] hat{phi}$. Which gives me $vec{R} = Rhat{R} + Rsintheta hat{phi}$ not $Rhat{R}$ now where i am misunderstanding or miscalculating ?






      vectors vector-analysis spherical-coordinates




      vectors vector-analysis spherical-coordinates






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      edited Jan 14 at 16:13









      mechanodroid

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      asked Jan 13 at 1:19









      user187604user187604

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          You made a mistake when calculating $hat{phi}$. We have



          $$frac{dhat{R}}{dphi} = -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}$$
          so
          $$hatphi = frac{frac{dhat{R}}{dphi}}{left|frac{dhat{R}}{dphi}right|} = frac{ -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}}{Rsintheta}= - sinphihat{i} + cosphihat{j}$$



          Now we have
          $$leftlangle vec{R}, hatphirightrangle = -Rsinthetacosphisinphi + Rsinthetasinphicosphi = 0$$
          which gives the correct result $vec{R} = Rhat{R}$.






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            $begingroup$

            You made a mistake when calculating $hat{phi}$. We have



            $$frac{dhat{R}}{dphi} = -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}$$
            so
            $$hatphi = frac{frac{dhat{R}}{dphi}}{left|frac{dhat{R}}{dphi}right|} = frac{ -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}}{Rsintheta}= - sinphihat{i} + cosphihat{j}$$



            Now we have
            $$leftlangle vec{R}, hatphirightrangle = -Rsinthetacosphisinphi + Rsinthetasinphicosphi = 0$$
            which gives the correct result $vec{R} = Rhat{R}$.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              You made a mistake when calculating $hat{phi}$. We have



              $$frac{dhat{R}}{dphi} = -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}$$
              so
              $$hatphi = frac{frac{dhat{R}}{dphi}}{left|frac{dhat{R}}{dphi}right|} = frac{ -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}}{Rsintheta}= - sinphihat{i} + cosphihat{j}$$



              Now we have
              $$leftlangle vec{R}, hatphirightrangle = -Rsinthetacosphisinphi + Rsinthetasinphicosphi = 0$$
              which gives the correct result $vec{R} = Rhat{R}$.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                You made a mistake when calculating $hat{phi}$. We have



                $$frac{dhat{R}}{dphi} = -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}$$
                so
                $$hatphi = frac{frac{dhat{R}}{dphi}}{left|frac{dhat{R}}{dphi}right|} = frac{ -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}}{Rsintheta}= - sinphihat{i} + cosphihat{j}$$



                Now we have
                $$leftlangle vec{R}, hatphirightrangle = -Rsinthetacosphisinphi + Rsinthetasinphicosphi = 0$$
                which gives the correct result $vec{R} = Rhat{R}$.






                share|cite|improve this answer









                $endgroup$



                You made a mistake when calculating $hat{phi}$. We have



                $$frac{dhat{R}}{dphi} = -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}$$
                so
                $$hatphi = frac{frac{dhat{R}}{dphi}}{left|frac{dhat{R}}{dphi}right|} = frac{ -Rsinthetasinphi hat{i} + Rsinthetacosphi hat{j}}{Rsintheta}= - sinphihat{i} + cosphihat{j}$$



                Now we have
                $$leftlangle vec{R}, hatphirightrangle = -Rsinthetacosphisinphi + Rsinthetasinphicosphi = 0$$
                which gives the correct result $vec{R} = Rhat{R}$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 14 at 16:11









                mechanodroidmechanodroid

                27.6k62447




                27.6k62447






























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