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r sin in .NET
1 r sin Scanning QR Code In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET framework applications. QR Code Creator In .NET Framework Using Barcode creator for VS .NET Control to generate, create QR Code image in .NET applications. Expressing the transformation relation in terms of the matrix, we have ea =
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Code128 Creation In None Using Barcode generation for Online Control to generate, create Code128 image in Online applications. Barcode Encoder In VB.NET Using Barcode creator for VS .NET Control to generate, create barcode image in .NET framework applications. In the case of spherical polar coordinates, the inverse matrix is given by 1 0 0 0 = 0 r 0 0 r sin UPC Code Encoder In Java Using Barcode creator for Java Control to generate, create UPC Symbol image in Java applications. Bar Code Generator In Java Using Barcode creation for Android Control to generate, create bar code image in Android applications. It is also possible to derive a transformation relationship for the basis one forms. Once again, we recall the form of the basis one forms when using a coordinate basis. In that case they are exact differentials: a = dx a The noncoordinate basis is related to the coordinate basis in the following way: Bar Code Creator In None Using Barcode printer for Office Excel Control to generate, create bar code image in Microsoft Excel applications. Barcode Creator In ObjectiveC Using Barcode creator for iPhone Control to generate, create barcode image in iPhone applications. a = a b dx b
In the case of the basis one forms, the components of the transformation matrix are given the label a b . To work this out for spherical coordinates, we consider a single term; i.e., = b dx b = r dr + d + d = r sin d We conclude that the only nonzero component is given by = r sin
Cartan s Structure Equations
It is a simple matter to show that the transformation matrix, which this time is denoted by a b , is given by 1 0 0 0 = 0 r 0 0 r sin This is just the inverse matrix we found when transforming the basis vectors. To express the coordinate basis one forms in terms of the noncoordinate basis one forms, we use the inverse of this matrix; i.e., dx a = 1 a b b
In the case of spherical polar coordinates, this matrix is given by 1 0 0 1 = r 0 0 0 0
1 a 1 r sin
These transformation matrices are related to those used with the basis vectors in the following way: a b
1 b A Note on Notation
Consider a set of coordinates x 0 , x 1 , x 2 , x 3 . Suppose that we are working in a coordinate basis, i.e., ea = / x a . In this case, the metric or line element is written as g = ds 2 = gab dx a dx b If we are working with an orthonormal tetrad, we write the metric in terms of the basis one forms. In other words, we write ds 2 = g = a b a b
Cartan s Structure Equations
In many cases, the inner product (as represented by a b ) is diagonal. If we have a b = diag (1, 1, 1, 1), then we can write the metric in the following way: ds 2 = g = 0 0 1 1 2 2 3 3 0 2 1 2 3 We will be using this frequently throughout the book in speci c examples. We now turn to the task of computing curvature using the orthonormal basis. This type of calculation is sometimes referred to by the name tetrad methods. The equations used to perform the calculations are Cartan s structure equations. Cartan s First Structure Equation and the Ricci Rotation Coef cients
The rst step in computing curvature, using the methods we are going to outline in this chapter, is to nd the curvature one forms and the Ricci rotation coef cients. The notation used for this method makes it look a bit more mathematically sophisticated than it really is. In fact you may nd it quite a bit less tedious than the straightforward methods used to nd the Christoffel symbols and Riemann tensor in the last chapter. The main thrust of this technique is given a set of basis one forms a , we wish a to calculate the derivatives d . These quantities satisfy Cartan s rst structure equation, which is d a = a b
b (5.9) Note that we are using hatted indices, which indicates we are working in the noncoordinate basis. The a b are called curvature one forms, and they can be written in terms of the basis one forms a as follows:

