Introduction to Differential Geometry with Tensor Applications. Группа авторов

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).

1.5 Linear Equations

      Let us consider n linear equations such that

(1.10a)

      where x¹, x², …. 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_1jA^1j = a a_1jA^2j = 0 and a_1jA^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 .

1.6 Results on Matrices and Determinants of Systems

      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 n². 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

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