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(x1, x2, … xn),
from calculus, we have
Example 1.8.4. (a) If apqxpxq = 0 for all values of the independent variables x1, x2, … xn and apq‘s are constant, show that aij + aji = 0.
(b) If apqrxpxqxr = 0 for all values of the independent variables x1, x2, … xn and apqr‘s are constant, show that akij + akji + aikj + ajki + aijk + ajik = 0.
Solution: Differentiating:
(1.12a)
with respect to xi
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
Solution: We have
Example 1.8.6. If
Solution: From above result
Example 1.8.7. If
(1.13a)
(1.13b)
The above result can be stated as
1.9 Exercises
1 1. Write out in full the following expression.
2 2. Expand the following using the summation convention.
3 3. Prove the following.
4 4. Show that for all values of independentvariables, x1, x2, … .xn, and where xp’s are constants.
5 5. Calculate
6 6. Using the relation , show that
7 7. Express each of the following sums using the summation convention:
8 8. Evaluate each of the following (range of indices 1 to n):9.
9 10. If yi are n independent functions of variables xi and zi are n independent functions of yi and if and then show that .
10 11. If and a−1 times the cofactor of in the determinant of show that
11 12. Prove that where
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