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Chapter 10: Q9P (page 534)

$V_{j}$ If role="math" localid="1659267226224" $V_{i} = g_{i j}, V^{j}$ is a contravariant vector and is a covariant vector, show that $U^{i} V_{j}$ is a $2^{n d}$ -rank mixed tensor. Hint:Write the transformation equations for U and V and multiply them.

Short Answer

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Step by step solution

01

Given information.

Co-variant and contra-variant components are defined.

02

Definition of covariance and contravariance.

The components of a vector relative to a tangent bundle basis are covariant in differential geometry if they change with the same linear transformation as the basis. If they change as a result of the inverse transformation, they are contravariant.

03

Write the definitions of the covariant and contravariant vectors.

Write the definitions of the vectors.

$U_{i}^{'} = \frac{鈭� x_{i}^{'}}{鈭� x_{k}} U^{k} V_{i}^{'} = \frac{鈭� x_{j}}{鈭� x_{l}^{'}} V_{l}$

04

Multiply the two equations.

Multiply the above equations.

$U_{i}^{'} V_{j}^{'} = \frac{鈭� x_{i}^{'}}{鈭� x_{k}} \frac{鈭� x_{j}}{鈭� x_{l}^{'}} U^{k} V_{l}$

It can be seen that this is a second-order mixed tensor.

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Most popular questions from this chapter

Show that the nine quantities $T_{ij} = (鈭� V_{i}) / (鈭� x_{J})$ (which are the Cartesian components of $鈭� V$ where V is a vector) satisfy the transformation equations $(2.14)$ for a Cartesian $2^{nd}$ -rank tensor. Show that they do not satisfy the general tensor transformation equations as in $(10.12)$ . Hint: Differentiate $(10.9)$ or $(10.10)$ partially with respect to, say, $x'_{k}$ . You should get the expected terms [as in $(10.12)$ ] plus some extra terms; these extraneous terms show that $(鈭� V_{i}) / (鈭� x_{J})$ is not a tensor under general transformations. Comment: It is possible to express the components of $鈭� V$ correctly in general coordinate systems by taking into account the variation of the basis vectors in length and direction.

The square matrix in equation $(10.3)$ is called the Jacobian matrix J; the determinant of this matrix is the Jacobian $J = \det J$ which we used in Chapter 5 , Section 4 to find volume elements in multiple integrals. (Note that as in Chapter 3, J represents a matrix; J in italics is its determinant.) For the transformation to spherical coordinates in localid="1659266126385" $(10.1)$ and $(10.2)$ show that $J = \det J = r^{2} \sin 胃$ . Recall that the spherical coordinate volume element is $r^{2} \sin 胃诲谤诲胃诲蠒$ . Hint: Find $J^{T} J$ and note that $\det (J^{T} J) = (\det J)^{2} .$

(a) Write the triple scalar productin $A 鈰� (B 脳 C)$ tensor form and show that it is equal to the determinant in Chapter 6, equation. $(3 .2)$ Hint: See. $(5 .5)$

(b) Write equation $(3 .2)$ of Chapter 6 in tensor form to show the equivalence of the various expressions for the triple scalar product. Hint: Change the dummy indices as needed.

Show by the quotient rule (Section 3 ) that $C_{ijkm}$ in $(7.5)$ is a $4^{th}$ -rank tensor.

$\frac{鈭� B}{鈭� t} = - 鈭� 脳 E$

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