By James Foster, J. David Nightingale

This textbook offers an exceptional advent to a subject that's tremendous effortless to get slowed down in. I took a one semester direction that used this article as an undergraduate, within which i presumed the publication was once in simple terms first rate, yet then whilst I took a gradute direction that used Carroll's Spacetime and Geometry is whilst i actually got here to understand the guidance this publication gave me (not that Carroll's ebook is undesirable, I simply would not suggest it for a primary reading). let alone the ebook is beautiful reasonable so far as physics texts cross.

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**Sample text**

The gradient of this scalar field has components 8a ¢ = 8~. That is, all its components are zero, except for the bth, which is equal to one. We therefore conclude that at each point P of M the gradient of the bth coordinate (regarded as a scalar field) is the bth basis vector e b for covariant vectors at P. These basis vectors define the cotangent space T p of M at P. What exactly is T p and how is it related to T p? The short answer to both these questions is that T p is the dual of Tp, but this needs some explanation.

I, §15). } and {e j } as if they were alternate bases for the same space , referring a given vector oX to one or the 48 1 Vector and tensor fields other, and distinguishing its components by referring to th em as eit her contravariant or covariant. This practice is not really correct, though is common in Euclid ean space, and has its origins in confusing (or identifying'' ) T p with T p. Euclidea n space has an inner product , t he usual dot product oX . l of vector algebra . This inner produ ct allows us to associate a covariant vector oX * in T p with a given cont ravariant vector oX in Ti.

The chain rule gives au j au i' au i' ax ' auj ax with similar expressions for auj jay and auj jaz. So . 46) where uf, is a short-hand for the partial derivative auj jau i' . 47) as the transformation formula for the covariant components. There are two routes to the inverse transformations. The first is to note that primed and unprimed quantities are on an equal footing, so that primed and unprimed suffixes can be swapped. 49) for transforming components. 49). 2). Matrix methods (as explained in Sec.