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With this identi cation, when we write x 1 we mean x, while x 2 means y and so forth. This identi cation is entirely general, and we can use it to represent another coordinate system such as cylindrical coordinates. What represents what will be made clear from the context. Also, it will be convenient to move back and forth between this representation and the ones you are used to. When using more than one coordinate system, or more importantly when considering transformations between coordinate system, we need a way to apply this representation to two different sets of coordinates. One way to do so is to put primes on the indices. As an example, suppose we are considering cartesian and spherical coordinates within the context of the same problem. If we label cartesian coordinates with (x, y, z) x 1 , x 2 , x 3 , then we add primes to the labels used for spherical coordinates and write (r, , ) x 1 , x 2 , x 3 We will label the components of a vector in the same way, with a raise index. In the current example, the components of a vector A in cartesian coordinates would be given by A = A1 , A2 , A3 While primed coordinates would represent the same vector in spherical coordinates A = A1 , A2 , A3 another useful notation change is to write partial derivatives in a succinct way. We write f = x f x or, using indices, we write a xa There are two reasons for writing things in this apparently obscure fashion. The rst is that when you do relativity a great deal of writing is required. Hey! Anything that can cut back on that is a good thing. But we will see that the
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placement of the indices (as shown in a ) will prove to be more enlightening and useful. Unfortunately, at this point we re not ready to say why, so you ll just have to take my word for it and just keep in mind what the shorthand symbols mean.
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In many cases, we are not going to work with speci c components of an objectlike vector, but will rather work with a general component. In this book we will label components of objectlike vectors with lowercase Latin letters. For example, we can refer to the vector A by Aa As we get involved with relativity, a vector will have space and time components (it will be a four vector). In that case, the time component of the vector will be labeled by the index 0, and so the components of a four vector V will be given by V = V 0, V 1, V 2, V 3 Vector addition can be described in the following way: A + B = A0 + B 0 , A1 + B 1 , A2 + B 2 , A3 + B 3 while scalar multiplication can be written as A = A0 , A1 , A2 , A3 Keep in mind that some authors prefer to use (1, 2, 3, 4) as their indices, using 4 to label time. We will stick to using 0 to label the time coordinate, however. Many authors prefer to use the following labeling convention. When all four components (space and time) in an expression are used, Greek letters are used for indices. So if an author writes T the indices , , can range over (0, 1, 2, 3). In this context, Latin indices are reserved for spatial components only, and so in the expression S ij
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