# Why do tensor indices differ in the Euler equation (fluid dynamics)

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.

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
$$\frac{\partial \rho}{\partial t} = \sum_{k=1}^3 \frac{\partial}{\partial x_k} (\rho v_k) = \nabla \cdot (\rho \vec v)$$
• @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. Jan 29, 2021 at 20:24