Introduction to Differential Geometry with Tensor Applications

Consider the determinant and let the element be a function of x 1, x 2… xn , etc. Let be the cofactor of of det a .

Then, the derivative of a with respect to x 1is given by

Therefore, in general, we can write .

Example 1.8.1.Write the terms contained in S = aijxixj taking n = 3.

Solution: Since the index i (or j) occurs both in subscript and superscript, we first sum on i from 1 to 3, then on j from 1 to 3.

Example 1.8.2.Express the sum of .

Solution: Here, the number of terms is 3 3= 27.

Since the index i ( or j or k ) occurs both in subscript and superscript, we first sum on i from 1 to 3, then on each term of its 3 terms we sum j from 1 to 3. This results in 9 terms. Then, on each of the 9 terms we sum k from 1 to 3, which results in 27 terms. Like the last example, we sum

Example 1.8.3.If f is a function of n variables xi , write the differential of f.

Solution: Since f = f ( x 1, x 2, … xn ),

from calculus, we have

Example 1.8.4.(a) If apqxpxq = 0 for all values of the independent variables x 1, x 2, … xn and apq ‘s are constant, show that aij + aji = 0.

(b) If apqrxpxqxr = 0 for all values of the independent variables x 1, x 2, … xn and apqr ‘s are constant, show that akij + akji + aikj + ajki + aijk + ajik = 0.

Solution: Differentiating:

(1.12a)

with respect to xi

(1.12b)

Differentiating (1.12b), with respect to xj , we get

(b) Differentiating

with respect to xi

Differentiating with respect to xj , we get

Differentiating in the same way, with respect to xk we get

Example 1.8.5.If is a double system such that , show that .

Solution: We have , taking determinant ,

Example 1.8.6.If is a double system such that , show that either or .

Solution: From above result

Example 1.8.7.If and , show that

(1.13a)