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Commutation Coef cients and Basis One Forms in .NET framework
Commutation Coef cients and Basis One Forms QR Code Reader In VS .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications. Drawing QR Code ISO/IEC18004 In .NET Using Barcode drawer for Visual Studio .NET Control to generate, create Quick Response Code image in VS .NET applications. It is also possible to determine whether or not a basis is holonomic by examining the basis one forms. A one form can be expanded in terms of a set of coordinate basis one forms a as = a a = a dx a In the same way that we can expand a vector in a different basis, we can also expand a one form in terms of a nonholonomic basis. Again using hats to denote the fact that we are working with a nonholonomic basis, we can write QR Scanner In VS .NET Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Generate Bar Code In Visual Studio .NET Using Barcode drawer for Visual Studio .NET Control to generate, create barcode image in .NET applications. = a a
Scan Bar Code In VS .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Draw QR Code JIS X 0510 In C# Using Barcode creation for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in .NET applications. Cartan s Structure Equations
QR Code 2d Barcode Encoder In .NET Using Barcode generator for ASP.NET Control to generate, create QR Code ISO/IEC18004 image in ASP.NET applications. Making QR In VB.NET Using Barcode drawer for .NET Control to generate, create QR Code JIS X 0510 image in VS .NET applications. Both expansions represent the same one form. Given a particular set of basis one forms, it may be desirable to determine if it is holonomic. Once again we do this by calculating the commutation coef cients, but by a different method. Given a set of basis one forms a , the commutation coef cients can be found by calculating d a . This quantity is related to commutation coef cients in the following way: 1 d a = Cbc a b c 2 Now recall that for a coordinate basis, the basis one forms are given by a = dx a In the previous chapter, we learned that the antisymmetry of the wedge product leads to the following result for an arbitrary p form : d (d ) = 0 This means that for a coordinate basis, d a = 0. For spherical polar coordinates, if we choose the nonholonomic basis, the basis one forms are given by r = dr, = r d , Matrix 2D Barcode Maker In Visual Studio .NET Using Barcode drawer for Visual Studio .NET Control to generate, create 2D Barcode image in Visual Studio .NET applications. Code 3 Of 9 Encoder In VS .NET Using Barcode creator for VS .NET Control to generate, create ANSI/AIM Code 39 image in .NET applications. (5.6) Barcode Maker In .NET Framework Using Barcode creator for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. Leitcode Encoder In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create Leitcode image in .NET applications. = r sin d
Data Matrix Printer In None Using Barcode drawer for Excel Control to generate, create Data Matrix image in Office Excel applications. 2D Barcode Maker In VB.NET Using Barcode printer for VS .NET Control to generate, create 2D Barcode image in .NET framework applications. (5.7) Code128 Recognizer In VB.NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. Print Data Matrix ECC200 In .NET Framework Using Barcode encoder for Reporting Service Control to generate, create ECC200 image in Reporting Service applications. Using (5.6), we can compute the commutation coef cients for this basis. For example, d = d (r sin d ) = sin dr d + r cos d d UPC Symbol Creation In ObjectiveC Using Barcode creation for iPhone Control to generate, create UPC Code image in iPhone applications. Data Matrix 2d Barcode Recognizer In VB.NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET framework applications. Using the de nitions given in (5.7), we can rewrite this expression in terms of the basis one forms. First, notice that 1 1 sin dr d = dr sin d = dr r sin d = r r r USS128 Scanner In Visual Basic .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Universal Product Code Version A Reader In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. For the second term, we nd r cos d d =
Cartan s Structure Equations
cos cos (r d ) r d = (r d ) r sin d r r sin cot = r
Putting these results together, we obtain 1 cot d = r + r r The antisymmetry of the wedge product means we can write this expression as 1 cot d = r + r r 1 1 1 r 1 r + = 2 r r 2 cot cot r r
Now we compare with (5.6) to read off the commutation coef cients. We nd C r = C r =
and C = C =
cot r
Remember, if the commutation coef cients vanish, then the basis is holonomic. As we mentioned at the beginning of the chapter, it is often convenient to do calculations using the orthonormal basis but we may need to express results in the coordinate basis. We now explore the techniques used to transform between the two. Transforming between Bases
We can work out a transformation law between the coordinate and noncoordinate basis vectors by using the coordinate components of the noncoordinate basis vectors. These components are denoted by (ea )b and known as the tetrad. The meaning of these components is the same as we would nd for any vector. In other words, we use them to expand a noncoordinate basis vector in terms of the basis vectors of a coordinate basis: ea = (ea )b eb (5.8) Cartan s Structure Equations
For example, we can expand the noncoordinate basis for spherical polar coordinates in terms of the coordinate basis vectors as follows: er = (er )b eb = (er )r er + (er ) e + (er ) e e = e e = e eb = e eb = e
er + e
e + e
er + e
e + e
Earlier we stated that the noncoordinate basis vectors were er = r , 1 e = , r e = 1 r sin Comparison with the above indicates that (er )r = 1 1 e = r 1 e = r sin All other components are zero. The components (ea )b can be used to construct a transformation matrix that we label a b as this matrix represents a transforma tion between the global coordinates and the local Lorentz frame of an observer. In the case of spherical polar coordinates, we have 1 0 0 1 = r 0 0 0 0

