Q: Berry Phase Computation Equation $(6.7.16)$ in the book Lectures on Quantum Mechanics (second edition) by Weinberg reads
$$\gamma_{n}[C]=n \iint_{A[C]} d A \frac{\mathbf{B} \cdot \mathbf{e}[\mathbf{B}]}{|\mathbf{B}|^{3}}\tag{6.7.16}$$
It is remarked that "the integral being taken over any area in the space of the magnetic field vector surrounded by the curve $C$. We can evaluate this integral using Gauss's theorem. Draw a cone (not a circular cone unless $C$ happens to be a circle) with base $A[C]$ and sides running from the origin in field space to the curve $C$. The integral (6.7.16) may be written as an integral over the whole surface of this cone, since on the sides of this cone the normal $\mathbf{e}$ is perpendicular to $\mathbf{B}$, and so these sides do not contribute to the surface integral."
I don't understand how the normal of the sides is necessarily perpendicular to $\mathbf{B}$, suppose, for convenience, we have a flat surface such that $\mathbf{B}$ is perpendicular to this surface, then since the sides cannot be perpendicular to the base, it follows that the normal of the sides cannot possibly be perpendicular to $\mathbf{B}$ right?
