Wedge procut)

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The concept of wedge product is introduced by the definition of surface in topology and its example.

Basic knowledge

First, let's look at the definition of the surface in Euclidean space:

All vertices W in omega (points in the parameter space) are recorded:
 
Corresponding to vertices in R3 (Euclidean space:

The matrix X _ {\ star} (the first-order partial derivative of W is arranged in a certain order) of W is defined as follows:

Using a matrix, we can define the split plane as a space consisting of X _ {u} (W) and X _ {v} (W), as shown below:

The external integral varpi is defined as follows (Because Omega is two-dimensional, the results of the wedge operator here mean two area vectors of the first-order reciprocal, that is, the result is the surface normal vector, while the model is area ):

The above proof is as follows:

 

We know that the absolute value of the second-order determinant is the area. Therefore, the 1 formula represents the area of the parallelogram of the adjacent edge by X _ {u} wedge X _ {v} and X _ {u} wedge X _ {v. Pass.

Correspondingly, the normal of the small plane where W points are located can be directly introduced by varpi, including:


Note that the preceding two formulas are similar to the vector product of a three-dimensional space. In fact, vector product, that is, cross multiplication, is only a special case of the outer product in three-dimensional space. In some analytic geometric documents, the vector product is called the outer product, which is easy to cause confusion. It can be said that in many cases, some people think that the outer product and the vector product are different expressions of a thing (area vector and normal vector, or vector and its geometric meaning ). At present, we can understand that the wedge product can be extended to any dimension space.

You can obtain the area A of the whole surface by integrating the tiny area at any point of interest on the surface.

 
Gaussian ing (Gauss map, normal map, or spherical map): ing three-dimensional surface Omega to two-dimensional unit sphere S2. Using the unit normal N above, there are:

Wedge procut)