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Chapter 10: Q10P (page 505)

$X_{i} A_{j} = B_{i j}$ .

Short Answer

Expert verified

Answer

The equation has been proven.

Step by step solution

01

Given Information

The tensor $X_{i} A_{j} = B_{i j}$ .

02

Definition of a cartesian tensor.

The first rank tensor is just a vector. A tensor of the second rank has nine components (in three dimensions) in every rectangular coordinate system.

03

Prove that X is a tensor.

Let $X_{i} A_{j} = B_{i j}$ where B is a non-zero tensor.

Apply transformation law on A and B.

0 = X'_{α} A'_{β} - B'_{α β} 0 = X'_{α} A'_{β} - a_{α i} a_{β j} - B'_{i j} 0 = X'_{α} A'_{β} - a_{α i} a_{β j} X_{i} A'_{j} 0 = X'_{α} A'_{β} - a_{α i} a_{β j} X_{i} A'_{γ} 0 = X'_{α} A'_{β} - a_{α i} δ_{β j} X_{i} A'_{γ} 0 = (X'_{α} - X_{i}) A'_{β}

A is an arbitrary vector.

Hence $(X'_{α} - a_{α i} X_{i}) = 0$

Thus, X is a tensor.

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Most popular questions from this chapter

Parabolic cylinder.

Write out the sums $P_{i j} e_{j}$ for each value of and compare the discussion of $(1 .1)$ .Hint: For example, if $i = 2$ [or y in $(1 .1)$ ], then the pressure across the face perpendicular to the $x_{2}$ axis is $P_{21} e_{1} + P_{22} e_{22} + P_{23} e_{3}$ , or, in the notation of (1.1), $P_{y x} i + P_{y y} j + P_{y z} k$ .

Find ${ds}^{2}$ in spherical coordinates by the method used to obtain(8.5)for cylindrical coordinates. Use your result to find for spherical coordinates, the scale factors, the vector ds , the volume element, the basis vectors $a_{r}, a_{θ}, a_{ϕ}$ and the corresponding unit basis vectors $e_{r}, e_{θ}, e_{ϕ}$ . Write the $g_{ij}$ matrix.

Consider the matrix A in $(2.7) o r (2.10)$ .Think of the elements in each row (or column) as the components of a vector. Show that the row vectors form an orthonormal triad (that is each is of unit length and they are all mutually orthogonal), and the column vectors form an orthonormal triad.

In equation $(10.13),$ let the variables be rectangular coordinates x, y, z, and let $x_{1}, x_{2}, x_{3}$ , be general curvilinear coordinates, orthogonal or not (see end of Section 8 ). Show that $J^{T} J$ is the $g_{ij}$ matrix in $(8.13)$ [or in $(8.16)$ for an orthogonal system]. Thus show that the volume element in a general coordinate system is $dV = \sqrt{g} {dx}_{1} {dx}_{2} {dx}_{3}$ where $g = {detg}_{ij}$ , and that for an orthogonal system, this becomes [by $(8.16)$ or $(10.19)$ ], $dV = h_{1} h_{2} h_{3} {dx}_{1} {dx}_{2} {dx}_{3}$ . Hint: To evaluate the products of partial derivatives in $J^{T} J$ , observe that the same expressions arise as in finding ${ds}^{2}$ . In fact, from $(8.11)$ and $(8.12)$ , you can show that row i times column j in $J^{T} J$ is just $a_{i} . a_{j} = g_{ij}$ in equations $(8.11)$ to $(8.14)$ .

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