5.4: Linear Independence in Terms of Metric Coefficients

The two partial derivative vectors are linearly dependent if and only if their cross product is zero. Thus the condition for regularity is that X_u(u,v) X_v( u,v) is not zero.

Recall that for any two vectors A and B , we have

|A B|²=|A|²|B| ²sin²(q)

|A B|²=|A|²|B| ²-|A| ²|B| ²cos²(q)
=|A| ²|B| ²-|A B|²

In the case of the partial derivative vectors of a vector function, this gives

|X_u( u,v) x X_v(u,v)|=g ₁₁(u,v)g₂₂( u,v)-g₁₂(u,v) ²

Our notation and this last formula suggest a matrix representation of what are formally called the metric coefficients :

(g_ij)=(g₁₁g₁₂g₂₁g₂₂)

Since the determinant of the metric coefficients appears in many different places it has its own name:

g(u,v)=det(g₁₁(u,v)g₁₂(u,v)g₂₁(u,v)g₂₂(u,v))