the normal vector

The elements of the normal vector are denoted as , a=1,...,dim. But for the moment, we use "norma" for , and "normb" for . First of all, we assume the following expressions of these values using unknown na2, na3, nb2, nb3.

The inner product of the normal vector and a tangent vector is foo7=.

The squared norm of the normal vector   is foo8= .

Now, we will solve na2, na3, nb2, nb3 from the equations foo7==0, foo8==1. First we get the coefficients in foo7 and foo8, and relabel nonfree indexes.

We get foo15= and foo16= below.

In the below, we confirm if the normal vector is orthogonal to the tangent vectors and if the length of the normal vector is 1.

The inner product of the normal vector and a tangent vector is foo17=.

The squared norm of the normal vector   is foo18= .