Introduction to Differential Geometry with Tensor Applications

Proof: We shall prove this result by taking the range of the indices from 1 to 2, but the results hold, in general, when they range from 1 to n .

We get . Hence, .

Taking the determinant of both sides, we get , as we know | AB | = | A || B |.

Property 1.6.2.If , then, and , where ( bik ) Tis the transpose of

Proof: We have , hence, .

Therefore,

Taking determinants of both sides, we get (since │ AT │ = │ A │).

Property 1.6.3.Let the cofactor of the element in the determinant be denoted by . Then, by summation convention we have

If the cofactor of aij is represented by Akj , it is expressed by the equation:

If we divide the cofactor Akj of the element of akj by the value a of the determinant, we form the normalized cofactor , represented by:

The above equation becomes

Property 1.6.4.Let us consider a system of n linear equations:

for n unknown xi , where

, where is cofactor of .

, which is called Cramer’s Rule , for the solution of n linear equations.

Property 1.6.5.Considering the transformation zi = zi ( yk ) and yi = yi ( xk ), let N function zi ( yk ) be of independent N variables of yk so that .

Here, N equation zi = zi ( yk ) is solvable for the z’s in terms terms of yi ’ s .

Similarly, yi = yi ( xk ) is a solution of yi in terms of xi ’ s so that .

Now, we have by the chain rule of differentiation that

Taking the determinant, we get

(1.11)

Considering a particular case in which zi = xi , Equation (1.5)becomes

Or

This implies that the Jacobian of Direct Transformation is the reciprocal of the Jacobian of Inverse Transformation.