proving the laplacian of a vector in cylindrical coordnates












3














I am proving the following identity for the laplacian of a vector $vec{v}=<v_r,v_theta,v_z>$ in cylindrical coordinates:
$$nabla^2 vec{v}=left( frac{partial^2 v_r}{partial r^2}+frac{1}{r^2}frac{partial^2 v_r}{partial theta^2}+frac{partial^2 v_r}{partial z^2}+frac{1}{r}frac{partial v_r}{partial r}-frac{2}{r^2}frac{partial v_theta}{partial theta} -frac{v_r}{r^2}right )vec{e_r} \ + left (frac{partial^2 v_theta}{partial r^2}+frac{1}{r^2}frac{partial^2 v_theta}{partial theta^2}+frac{partial^2 v_theta}{partial z^2}+frac{1}{r}frac{partial v_theta}{partial r}+frac{2}{r^2}frac{partial v_r}{partial theta}-frac{v_theta}{r^2} right )vec{e_theta} \ left( frac{partial^2 v_z}{partial r^2}+frac{1}{r^2}frac{partial^2 v_z}{partial theta^2}+frac{1}{r}frac{partial v_z}{partial r}+frac{partial^2 v_z}{partial z^2} right)vec{e_z}$$ I am able to derive the following identity for the Laplacian operator in cylindrical coordinates $$nabla^2=frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} $$. So to prove the desired identity, $$nabla^2 vec{v}=left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_rvec{e_r}+v_theta vec{e_theta}+v_zvec{e_z}) \
= left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_rvec{e_r})+ left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_thetavec{e_theta})+ left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_zvec{e_z})$$. And upon distributing the vector components to the operator I finally get $$nabla^2 vec{v}=left( frac{partial^2 v_r}{partial r^2}+frac{1}{r}frac{partial v_r}{partial r}+frac{1}{r^2}frac{partial^2 v_r}{partial theta^2}+frac{partial^2 v_r}{partial z^2}-frac{v_r}{r^2} right)vec{e_r} \
+left( frac{partial^2 v_theta}{partial r^2}+frac{1}{r}frac{partial v_theta}{partial r}+frac{1}{r^2}frac{partial^2 v_theta}{partial theta^2}-frac{v_theta}{r^2}+frac{partial^2 v_theta}{partial z^2} right)vec{e_theta} \
left( frac{partial^2 v_z}{partial r^2}+frac{1}{r}frac{partial v_z}{partial r}+frac{1}{r^2}frac{partial^2 v_z}{partial theta^2}+frac{partial^2 v_z}{partial z^2} right)vec{e_z}$$ which is not the same with the identity. I am confused how the $-frac{2}{r^2}frac{partial v_theta}{partial theta}$ and $frac{2}{r^2}frac{partial v_r}{partial theta}$ appeared in the $vec{e_r}$ and $vec{e_theta}$ components, respectively. Where did I go wrong? Need help...thanks






vector-analysis







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  • 3




    The vectors $e_r, e_z$ and $e_theta$ are not constant when $r, theta, z$ vary. (This is a distinctive trait of Cartesian coordinates). You should differentiate those vectors as well. mathworld.wolfram.com/CylindricalCoordinates.html
    – Giuseppe Negro
    Oct 5 '15 at 16:23
















3














I am proving the following identity for the laplacian of a vector $vec{v}=<v_r,v_theta,v_z>$ in cylindrical coordinates:
$$nabla^2 vec{v}=left( frac{partial^2 v_r}{partial r^2}+frac{1}{r^2}frac{partial^2 v_r}{partial theta^2}+frac{partial^2 v_r}{partial z^2}+frac{1}{r}frac{partial v_r}{partial r}-frac{2}{r^2}frac{partial v_theta}{partial theta} -frac{v_r}{r^2}right )vec{e_r} \ + left (frac{partial^2 v_theta}{partial r^2}+frac{1}{r^2}frac{partial^2 v_theta}{partial theta^2}+frac{partial^2 v_theta}{partial z^2}+frac{1}{r}frac{partial v_theta}{partial r}+frac{2}{r^2}frac{partial v_r}{partial theta}-frac{v_theta}{r^2} right )vec{e_theta} \ left( frac{partial^2 v_z}{partial r^2}+frac{1}{r^2}frac{partial^2 v_z}{partial theta^2}+frac{1}{r}frac{partial v_z}{partial r}+frac{partial^2 v_z}{partial z^2} right)vec{e_z}$$ I am able to derive the following identity for the Laplacian operator in cylindrical coordinates $$nabla^2=frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} $$. So to prove the desired identity, $$nabla^2 vec{v}=left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_rvec{e_r}+v_theta vec{e_theta}+v_zvec{e_z}) \
= left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_rvec{e_r})+ left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_thetavec{e_theta})+ left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_zvec{e_z})$$. And upon distributing the vector components to the operator I finally get $$nabla^2 vec{v}=left( frac{partial^2 v_r}{partial r^2}+frac{1}{r}frac{partial v_r}{partial r}+frac{1}{r^2}frac{partial^2 v_r}{partial theta^2}+frac{partial^2 v_r}{partial z^2}-frac{v_r}{r^2} right)vec{e_r} \
+left( frac{partial^2 v_theta}{partial r^2}+frac{1}{r}frac{partial v_theta}{partial r}+frac{1}{r^2}frac{partial^2 v_theta}{partial theta^2}-frac{v_theta}{r^2}+frac{partial^2 v_theta}{partial z^2} right)vec{e_theta} \
left( frac{partial^2 v_z}{partial r^2}+frac{1}{r}frac{partial v_z}{partial r}+frac{1}{r^2}frac{partial^2 v_z}{partial theta^2}+frac{partial^2 v_z}{partial z^2} right)vec{e_z}$$ which is not the same with the identity. I am confused how the $-frac{2}{r^2}frac{partial v_theta}{partial theta}$ and $frac{2}{r^2}frac{partial v_r}{partial theta}$ appeared in the $vec{e_r}$ and $vec{e_theta}$ components, respectively. Where did I go wrong? Need help...thanks






vector-analysis







share|cite|improve this question


















  • 3




    The vectors $e_r, e_z$ and $e_theta$ are not constant when $r, theta, z$ vary. (This is a distinctive trait of Cartesian coordinates). You should differentiate those vectors as well. mathworld.wolfram.com/CylindricalCoordinates.html
    – Giuseppe Negro
    Oct 5 '15 at 16:23














3












3








3


3





I am proving the following identity for the laplacian of a vector $vec{v}=<v_r,v_theta,v_z>$ in cylindrical coordinates:
$$nabla^2 vec{v}=left( frac{partial^2 v_r}{partial r^2}+frac{1}{r^2}frac{partial^2 v_r}{partial theta^2}+frac{partial^2 v_r}{partial z^2}+frac{1}{r}frac{partial v_r}{partial r}-frac{2}{r^2}frac{partial v_theta}{partial theta} -frac{v_r}{r^2}right )vec{e_r} \ + left (frac{partial^2 v_theta}{partial r^2}+frac{1}{r^2}frac{partial^2 v_theta}{partial theta^2}+frac{partial^2 v_theta}{partial z^2}+frac{1}{r}frac{partial v_theta}{partial r}+frac{2}{r^2}frac{partial v_r}{partial theta}-frac{v_theta}{r^2} right )vec{e_theta} \ left( frac{partial^2 v_z}{partial r^2}+frac{1}{r^2}frac{partial^2 v_z}{partial theta^2}+frac{1}{r}frac{partial v_z}{partial r}+frac{partial^2 v_z}{partial z^2} right)vec{e_z}$$ I am able to derive the following identity for the Laplacian operator in cylindrical coordinates $$nabla^2=frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} $$. So to prove the desired identity, $$nabla^2 vec{v}=left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_rvec{e_r}+v_theta vec{e_theta}+v_zvec{e_z}) \
= left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_rvec{e_r})+ left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_thetavec{e_theta})+ left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_zvec{e_z})$$. And upon distributing the vector components to the operator I finally get $$nabla^2 vec{v}=left( frac{partial^2 v_r}{partial r^2}+frac{1}{r}frac{partial v_r}{partial r}+frac{1}{r^2}frac{partial^2 v_r}{partial theta^2}+frac{partial^2 v_r}{partial z^2}-frac{v_r}{r^2} right)vec{e_r} \
+left( frac{partial^2 v_theta}{partial r^2}+frac{1}{r}frac{partial v_theta}{partial r}+frac{1}{r^2}frac{partial^2 v_theta}{partial theta^2}-frac{v_theta}{r^2}+frac{partial^2 v_theta}{partial z^2} right)vec{e_theta} \
left( frac{partial^2 v_z}{partial r^2}+frac{1}{r}frac{partial v_z}{partial r}+frac{1}{r^2}frac{partial^2 v_z}{partial theta^2}+frac{partial^2 v_z}{partial z^2} right)vec{e_z}$$ which is not the same with the identity. I am confused how the $-frac{2}{r^2}frac{partial v_theta}{partial theta}$ and $frac{2}{r^2}frac{partial v_r}{partial theta}$ appeared in the $vec{e_r}$ and $vec{e_theta}$ components, respectively. Where did I go wrong? Need help...thanks






vector-analysis







share|cite|improve this question













I am proving the following identity for the laplacian of a vector $vec{v}=<v_r,v_theta,v_z>$ in cylindrical coordinates:
$$nabla^2 vec{v}=left( frac{partial^2 v_r}{partial r^2}+frac{1}{r^2}frac{partial^2 v_r}{partial theta^2}+frac{partial^2 v_r}{partial z^2}+frac{1}{r}frac{partial v_r}{partial r}-frac{2}{r^2}frac{partial v_theta}{partial theta} -frac{v_r}{r^2}right )vec{e_r} \ + left (frac{partial^2 v_theta}{partial r^2}+frac{1}{r^2}frac{partial^2 v_theta}{partial theta^2}+frac{partial^2 v_theta}{partial z^2}+frac{1}{r}frac{partial v_theta}{partial r}+frac{2}{r^2}frac{partial v_r}{partial theta}-frac{v_theta}{r^2} right )vec{e_theta} \ left( frac{partial^2 v_z}{partial r^2}+frac{1}{r^2}frac{partial^2 v_z}{partial theta^2}+frac{1}{r}frac{partial v_z}{partial r}+frac{partial^2 v_z}{partial z^2} right)vec{e_z}$$ I am able to derive the following identity for the Laplacian operator in cylindrical coordinates $$nabla^2=frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} $$. So to prove the desired identity, $$nabla^2 vec{v}=left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_rvec{e_r}+v_theta vec{e_theta}+v_zvec{e_z}) \
= left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_rvec{e_r})+ left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_thetavec{e_theta})+ left(frac{partial^2}{partial r^2}+frac{1}{r}frac{partial}{partial r}+frac{1}{r^2}frac{partial^2}{partial theta^2}+frac{partial^2}{z^2} right)(v_zvec{e_z})$$. And upon distributing the vector components to the operator I finally get $$nabla^2 vec{v}=left( frac{partial^2 v_r}{partial r^2}+frac{1}{r}frac{partial v_r}{partial r}+frac{1}{r^2}frac{partial^2 v_r}{partial theta^2}+frac{partial^2 v_r}{partial z^2}-frac{v_r}{r^2} right)vec{e_r} \
+left( frac{partial^2 v_theta}{partial r^2}+frac{1}{r}frac{partial v_theta}{partial r}+frac{1}{r^2}frac{partial^2 v_theta}{partial theta^2}-frac{v_theta}{r^2}+frac{partial^2 v_theta}{partial z^2} right)vec{e_theta} \
left( frac{partial^2 v_z}{partial r^2}+frac{1}{r}frac{partial v_z}{partial r}+frac{1}{r^2}frac{partial^2 v_z}{partial theta^2}+frac{partial^2 v_z}{partial z^2} right)vec{e_z}$$ which is not the same with the identity. I am confused how the $-frac{2}{r^2}frac{partial v_theta}{partial theta}$ and $frac{2}{r^2}frac{partial v_r}{partial theta}$ appeared in the $vec{e_r}$ and $vec{e_theta}$ components, respectively. Where did I go wrong? Need help...thanks






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asked Feb 24 '15 at 18:41









james25

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  • 3




    The vectors $e_r, e_z$ and $e_theta$ are not constant when $r, theta, z$ vary. (This is a distinctive trait of Cartesian coordinates). You should differentiate those vectors as well. mathworld.wolfram.com/CylindricalCoordinates.html
    – Giuseppe Negro
    Oct 5 '15 at 16:23














  • 3




    The vectors $e_r, e_z$ and $e_theta$ are not constant when $r, theta, z$ vary. (This is a distinctive trait of Cartesian coordinates). You should differentiate those vectors as well. mathworld.wolfram.com/CylindricalCoordinates.html
    – Giuseppe Negro
    Oct 5 '15 at 16:23








3




3




The vectors $e_r, e_z$ and $e_theta$ are not constant when $r, theta, z$ vary. (This is a distinctive trait of Cartesian coordinates). You should differentiate those vectors as well. mathworld.wolfram.com/CylindricalCoordinates.html
– Giuseppe Negro
Oct 5 '15 at 16:23




The vectors $e_r, e_z$ and $e_theta$ are not constant when $r, theta, z$ vary. (This is a distinctive trait of Cartesian coordinates). You should differentiate those vectors as well. mathworld.wolfram.com/CylindricalCoordinates.html
– Giuseppe Negro
Oct 5 '15 at 16:23










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The product rule for second order differentiation is $(fg)'' = f''g + 2f'g'+ fg''$. You simply omitted the middle value.






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    The product rule for second order differentiation is $(fg)'' = f''g + 2f'g'+ fg''$. You simply omitted the middle value.






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      The product rule for second order differentiation is $(fg)'' = f''g + 2f'g'+ fg''$. You simply omitted the middle value.






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        The product rule for second order differentiation is $(fg)'' = f''g + 2f'g'+ fg''$. You simply omitted the middle value.






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        The product rule for second order differentiation is $(fg)'' = f''g + 2f'g'+ fg''$. You simply omitted the middle value.







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        edited Nov 11 '17 at 16:53









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        answered Nov 11 '17 at 16:17









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