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I am working through the Landau Lifshitz book on Hydrodynamics (sorry, I only have the Russian version). In the chapter where the momentum tensor is derived, the equations are presented in tensor notation. As you can see in the picture, (1) is the momentum equation, (2) is the continuity equation, and (3) is the Euler equation.

Could someone provide an explanation why the tensor indices differ and what does that mean, please? Eq. (3), for instance, has both i and k indices.

enter image description here

Thank you for your help!

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The first and third equations are shorthand for three equations each, one for each $i=1,2,3$. When an index appears twice (as $k$ does in equations 2 and 3), it means that index is summed over. For instance, equation 2 reads

$$\frac{\partial \rho}{\partial t} = \sum_{k=1}^3 \frac{\partial}{\partial x_k} (\rho v_k) = \nabla \cdot (\rho \vec v)$$

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  • $\begingroup$ Thank you very much. Now, this makes sense. Just a short follow-up question: In (3) on the right-hand side, do I first expand the shorthand into a summation and then multiply each of the terms times v_i ? $\endgroup$
    – euler132
    Jan 29, 2021 at 20:03
  • $\begingroup$ @euler132 The first term on the right hand side of equation 3 is $\sum_{k=1}^3 v_k \frac{\partial v_i}{\partial x_k} = \vec v \cdot (\nabla v_i)$ in vector notation. $\endgroup$
    – J. Murray
    Jan 29, 2021 at 20:24

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