00082___0acd08daa6f30563f3a6c44826985709 - tensor with...

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Sec. 2.16 PHYSICAL COMPONENTS OF A VECTOR 61 not have the same physical dimensions. This difficulty (and it is also a great convenience!) arises because we would like to keep our freedom in choosing arbitrary curvilinear coordinates. Thus, in spherical polar coordinates for a Euclidean space, the position of a point is expressed by a length and two angles. In a four-dimensional space a point may be expressed in three lengths and a time. For this reason, we must distinguish the tensor compo- nents from the "physical components," which must have uniform physical dimensions. PROBLEMS 2.12. Show that in an orthogonal n-dimensional coordinates system, we have for each component i = j, 9' = l/gij. If aij is a tensor, and the components aij = aj,, then the tensor aij is called a symmetric tensor. If the components ai, = -aj,, then the tensor aij is said to be skew-symmetric, or antisymmetric. Show that the symmetry of a
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Unformatted text preview: tensor with respect to two indices at the same level is conserved under coordi- nate tmnsfonations. Since aij = !j(aij + aji) + !j(aij - aji), any covariant (or contmvariant) second-order tensor can be written as the sum of a symmetric and a skew-symmetric tensor. Show that the scalar product of a symmetric tensor Sii and a skew- symmetric tensor Wij vanishes indentically. 2.16. Show that the Cartesian tensor Wik = eijkuj is skew-symmetric, where uj is a vector. 2.16. Show that if A j k is a skew-symmetric Cartesian tensor, then the unique solution of the equation wi = ieijkAjk is Amn = emniWi. Let V be the operator 2.13. 2.14. 2.17. d v = g r - - . aer Show that gradq5=V+=gr-, 84 aer divF = V .F = F'I,, Show that these functions are invariant under coordinate transformations. T+!J(Z~,Z~,Z~) be a scalar field. Show that 2.18. Let gap be the associated contravariant Euclidean metric tensor and (a) g"p$lap is a scalar field....
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