Introduction to Differential Geometry with Tensor Applications

(1.7)

Property 1.4.5.

Also, by definition,

In particular, when i = k , we get

Remark 1.4.1.If we multiply xk by , we simply replace index k of xk with index i and for this reason, is called a substitution factor .

Example 1.4.1.Evaluate (a) and (b) where the indices take all values from 1 to n.

(1.8a)

(1.8b)

(b) by 1.8b

Example 1.4.2.If xi and yi are independent coordinates of a point, it is shown that

Solution: The partial derivative of ϕ in two coordinate systems are different and are connected by the following formula of Differential Calculus:

(1.9a)

Since xj is independent of, when j ≠ i

(1.9b)

Hence, the result follows from (1.9a)and (1.9b).

Let us consider n linear equations such that

(1.10a)

where x 1, x 2, …. xn are n unknown variables.

Let us consider:

For the expansion of det | ai j | in terms of cofactors we have

(1.10b)

where a = | ai j | and the cofactor of ai j is Ai j .

We can derive Cramer’s Rule for the solution of the system of n linear equations:

Now, multiplying both sides of (1.10a)by Ai j , we get

by (1.10b), we get, axj = biAi j .

From here, we can easily get

Example 1.5.1.Show that , where a is a determinant ai jie a = | ai j | of order 3 and Ai j are cofactors of ai j .

Solution: By expansion of determinants, we have:

Which can be written as a 1j A 1j= a a 1j A 2j= 0 and a 1j A 3j= 0 [we know aijAij = a ].

Similarly, we have

Using Kronecker Delta Notation, these can be combined into a single equation:

All nine of these equations can be combined into .

It is known that if the range of the indices of a system of second order are from 1 to n , the number of components is n 2. Systems of second order are organized into three types: ai j , ai j , and their matrices,

each of which is an n × n matrix.

We shall now establish the following results:

Property 1.6.1.If , then and .