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Let the components of be , those of be and those of be . Find (i) ; (ii) ; (iii) ; (iv) .
(2a) show that they are linearly independent.
(2b) Find the components of if
(2c) Find the value of for
(2d) Determine whether the one- forms , , and are linearly independent if , , , , , , , .
find (i) the components of the symmetric tensor and the antisymmetric tensor ; (ii) the components of ; (iii) the components of ; and (iv) the components of .
(4b) For the tensor whose components are , does it make sense to speak of its symmetric and antisymmetric parts? If so, define them. If not, say why.
(4c) Raise an index of the metric tensor to prove that
and the scalar has the value
(6a) Find , , . Is suitable as a 4- velocity field? Is ?
(6b) Find the spatial velocity of a particle whose four- velocity is , for arbitrary t. What happens to it in the limits , ?
(6c) Find for all .
(6d) Find for all and .
(6e) Show that for all . Show that for all .
(6f) Find .
(6g) Find for all .
(6h) Find and compare with (f) above. Why are the two answers similar?
(6i) Find for all . Find for all . What are the numbers the components of?
(6j) Find , . and .
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