# Understanding intuition behind integral involving mixed product

The topological charge i.e. skyrmion number (also called wrapping number) is defined as the number of times the spin vectors in a 2D configuration (i.e. lying on a 2D plane, as shown in the image above, bottom 2) wrap around a unit sphere (as shown in the image above, top 2). The integration is performed over this 2D configuration/plane (far away, the spins are zero). For a magnetic skyrmion, this is an integer number of times.

The topological charge is defined as follows:

$$Q = \frac{1}{4\pi} \int \vec{n} \cdot \left( \frac{\partial \vec{n}}{\partial x} \times \frac{\partial \vec{n}}{\partial y} \right) dxdy,$$

where $$\vec{n}$$ is the (normalized) spin vector. You can write the spin vector as follows:

$$\vec{n} = \begin{pmatrix} \sin\theta \cos\phi \\ \sin\theta \sin\phi \\ \cos\theta \end{pmatrix}$$

If I understand correctly, this can be seen as a mapping from a two-dimensional space to the surface of a sphere $$S^2$$, but not sure how to precisely define this.

Any way, I am curious behind the intuition of this integral. Why exactly does this integral calculate the number of times the spins in a 2D configuration wrap a unit sphere?

I recognize the fact that there is a mixed product which essentially measures the volume of the parallelepiped by the three vectors, and the fact that this could be related to solid angle, but it is not clear to me exactly how this follows (both mathematically and visually).

• What is $\vec n$ here? Normally it is used for the normal vector to the surface you are integrating over. But if that were the case, there would no dependence on the spin vectors it is supposed to measure, but would instead be a constant of the surface. And what do you mean by a "2D configuration"? Maybe your course defined this terminology, but it is not a generic concept you can just expect everyone to know. Commented May 24 at 1:05
• I edited it now to the post. Commented May 24 at 7:25
• The only thing you've clarified is that you are not sure what the terms you are tossing around mean mathematically. This is the real issue behind your question. Because you have only vague notions and no concept of how to adequately express them mathematically, you cannot understand the definition. But translating the physics into mathematics is a physics problem, not a mathematics problem, so you are less likely to get useful help in this mathematics forum. You need to clarify in your own head what "spin vectors", "2D configurations", "skyrmions" are mathematically. Commented May 26 at 3:10
• No, you haven't defined them. You've thrown around vague descriptions, and what you have given was in terms of other symbols $\theta, \phi$ that you failed to define. How are $x, y$ related to $\theta, \phi$? I could interpret $\theta,\phi$ as latitude/longitude on the sphere, but if I do, the integral becomes a constant. The problem is only mathematical once it is defined mathematically. It has not been. Until you have adequately translated the physics into mathematics, there is no mathematics to understand. Commented May 27 at 3:00
• Does this answer your question? Why is $\frac{1}{4\pi} \int \int_S dx \, dy\, \vec{n} \cdot ( \partial_x \vec{n} \times \partial_y \vec{n} ) = 1$? Commented Jun 27 at 5:48