Introduction to Differential Geometry with Tensor Applications - Группа авторов

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of n linear equations.

      Property 1.6.5. Considering the transformation zi = zi(yk) and yi = yi(xk), let N function zi(yk) be of independent N variables of yk so that image.

      Here, N equation zi = zi(yk) is solvable for the z’s in terms terms of yi’s.

      Now, we have by the chain rule of differentiation that

image

      Taking the determinant, we get

      (1.11) image

image

      Or

image

      This implies that the Jacobian of Direct Transformation is the reciprocal of the Jacobian of Inverse Transformation.

      Consider the determinant image and let the element image be a function of x1, x2 … xn, etc. Let image be the cofactor of image of det a.

      Then, the derivative of a with respect to x1 is given by

image

      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.

image

      Example 1.8.2. Express the sum of image.

      Solution: Here, the number of terms is 33 = 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

image

      Solution: Since f = f(x1, x2, … xn),

      from calculus, we have image

      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) image

      with respect to xi

image

      (b) Differentiating

image

      with respect to xi

image

      Differentiating with respect to xj, we get

image

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

image

      Solution: We have image, taking determinant image,

image

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

      Solution: From above result image

image