Introduction to Differential Geometry with Tensor Applications. Группа авторов

      The above result can be stated as . It is the result of the multiplication of two determinants of the third order.

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