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The force exerted by an electric field on a dipole is :

$$(\vec{p}.\vec{\nabla})\vec{E}$$

but how exactly do I develop this ?

Is it :

$$p_x\frac{\partial E}{\partial x}\vec{e_x} + p_y\frac{\partial E}{\partial y}\vec{e_y} + p_z\frac{\partial E}{\partial z} \vec{e_z}$$

in cartesian coordinates ?

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    $\begingroup$ $p_x\frac{\partial}{\partial x}\vec{E}+p_y\frac{\partial}{\partial y}\vec{E}+p_z\frac{\partial}{\partial z}\vec{E}$ $\endgroup$
    – lemon
    Commented Mar 12, 2015 at 10:40
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    $\begingroup$ $p^i \partial_i E_j$. Tensor calculus without index notation is a nightmare. $\endgroup$
    – Robin Ekman
    Commented Mar 12, 2015 at 10:51

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You're close: we can write the "operator" $p.\nabla$ as the "scalar" $$p.\nabla = p_x{\partial\over\partial x}+p_y{\partial\over\partial y}+p_z{\partial\over\partial z}$$

and $$E = E_x i + E_y j + E_z k = (E_x,E_y,E_z)^T$$

Then the result will be the vector $$\left(p_x{\partial E_x\over\partial x}+p_y{\partial E_x\over\partial y}+p_z{\partial E_x\over\partial z}, p_x{\partial E_y\over\partial x}+p_y{\partial E_y\over\partial y}+p_z{\partial E_y\over\partial z},p_x{\partial E_z\over\partial x}+p_y{\partial E_z\over\partial y}+p_z{\partial E_z\over\partial z}\right)^T$$

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