By Serge Lang
This 5th variation of Lang's booklet covers the entire themes typically taught within the first-year calculus series. Divided into 5 elements, each one component to a primary path IN CALCULUS includes examples and functions on the subject of the subject coated. additionally, the rear of the booklet comprises specific options to quite a few the routines, letting them be used as worked-out examples -- one of many major advancements over earlier versions.
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Additional info for A first course in calculus, Fifth Edition
2. precisely the original is U tensor It is easy to see that this of lowering or processes the tensor f. 213. Tnr, - on6(7s, E*ercisc. Find the mixed metric tensor oio, obtained from o^o by raising the second subscript. Before we introduced the metric tensor of Riemannian space we recognizrd a tensor as a geometrical object-a thing which had different representations in different coordinate systems, but at the same time an existence of its own. r and I-, were entirely unrelated; one was contravariant and the other conariant, and there was no connection between the one and the other.
The rule for differentiating a product, together with the fact that the covariant derivative of the metric tensor vanishes, implies that the order of the operations of lowering (or raising) a suffix and of differentiating can be reversed. 529. -F. This means that it is immaterial whether we differentiate the covariant, contravariant or one of the mixed representations of a tensor. Erqcisc. 530. Prove that d. 7;(a^X-Io) : elr 2o*nlr*fi , where )r'is any vector field given along a curye for which s is the arc length.
TNraw Specn dicator of the direction of the geodesic(. : + 1). 444 has the value zrcroitjre curve has a null direction at each of its points. Such a curve is a geodesicnull line. 44s. duzT\*n r \ryy:g. 446. 446. 44g. dr- dxn o^o66: o. By suitable choice of the parameter u, )r can be made any pr€assigned function of u. 449. dzr' . 448 hold. s GnopssrcNutt, Lnrs 47 Excrcisc. 445, are obtained from any one such parameter by the linear transformation ou*b, a: arbitrary constants. o and D being Like a geodesic, a geodesic null line is determined by a point on it, and the direction of its tangent there; but of course tlris direction must be a null direction.
A first course in calculus, Fifth Edition by Serge Lang