The differential acceleration $d a=g(x)-g(x-d x)$ is derived by performing a Taylor expansion of the gravitational field $g$ around the position $x$, retaining only the first-order terms in the small displacement $d x$.

In Cartesian coordinates, the $i$-th component of the gravitational field is $g^i=-\frac{\partial \phi}{\partial x}$. The Taylor expansion of $g^i(x-d x)$ is:

$$ g^i(x-d x) \approx g^i(x)-\frac{\partial g^i}{\partial x^j} d x^j $$

The differential acceleration is then:

$$ d a^i=g^i(x)-g^i(x-d x) \approx g^i(x)-\left[g^i(x)-\frac{\partial g^i}{\partial x^j} d x^j\right]=\frac{\partial g^i}{\partial x^j} d x^j $$

Thus, the tidal tensor is $T_j^i=\frac{\partial g^i}{\partial x^j}$.

How is the differential acceleration in a gravitational field derived using Taylor expansion-L.mp4