Solution

In the previous problem, we already showed that if the field divergence is zero, then its flow through any closed surface is zero.

Let's write down the Biot-Savare-Laplace law:

$$\hat{B} = \mu_0 I[d\hat{l} \times \frac{\hat{r}}{r^3}]$$

Note that the gradient from the following expression is:
$$\nabla\frac{1}{r} = -\frac{\hat{r}}{r^3}$$

Now let's find the divergence:
$$div \hat{B} =-\nabla\mu_0 I[d\hat{l} \times \nabla \frac{1}{r}]$$

Only $r$ depends on $x, y, z$, here we need one fact from mathematics, the rotor from the gradient is 0 $\to$ divergence is zero, which means the statement is proven.