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Nonholonomic Bases in Visual Studio .NET
Nonholonomic Bases QR Code Scanner In VS .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in VS .NET applications. Printing QRCode In .NET Framework Using Barcode creation for VS .NET Control to generate, create Denso QR Bar Code image in VS .NET applications. A nonholonomic basis is one such that the basis vectors are orthonormal with respect to the chosen metric. Another name for this type of basis is a noncoordinate basis and you will often hear the term orthonormal tetrad (more below). This type of basis is based on the fundamental ideas you are used to from freshman physics. A set of orthogonal vectors, each of unit length, are chosen for the basis. We indicate that we are working with an orthonormal basis by placing a hat or carat over the indices; i.e., basis vectors and basis one forms are written as QRCode Scanner In VS .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET framework applications. Encoding Barcode In .NET Framework Using Barcode generator for VS .NET Control to generate, create bar code image in .NET framework applications. ea , a
Scan Bar Code In VS .NET Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Encode QRCode In C#.NET Using Barcode generation for VS .NET Control to generate, create QR Code image in .NET framework applications. An orthonormal basis is of interest physically and has use beyond mere mathematics. Such a basis is used by a physical observer and represents a basis with respect to the local Lorentz frame, while the coordinate basis represents Create QR Code In .NET Using Barcode creation for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. Drawing QR Code In VB.NET Using Barcode maker for .NET framework Control to generate, create QR image in .NET applications. Cartan s Structure Equations
UPC  13 Creator In .NET Framework Using Barcode maker for .NET framework Control to generate, create EAN13 image in VS .NET applications. UCC.EAN  128 Maker In .NET Framework Using Barcode drawer for Visual Studio .NET Control to generate, create GTIN  128 image in Visual Studio .NET applications. the global spacetime. As we move ahead in this chapter, we will learn how to transform between the two representations. We can expand any vector V in terms of a coordinate or a noncoordinate basis. Just like any expansion, in terms of basis vectors, these are just different representations of the same vector Barcode Maker In .NET Using Barcode generator for .NET Control to generate, create barcode image in VS .NET applications. USPS Confirm Service Barcode Drawer In VS .NET Using Barcode creation for Visual Studio .NET Control to generate, create Planet image in .NET framework applications. V = V a ea = V a ea
Make USS Code 128 In Java Using Barcode generator for Java Control to generate, create ANSI/AIM Code 128 image in Java applications. DataMatrix Creation In Java Using Barcode drawer for Android Control to generate, create Data Matrix ECC200 image in Android applications. Since this basis represents the frame of the local Lorentz observer, we can use the at space metric to raise and lower indices in that frame. As usual, the signs of the components can be read off the metric. For example, with a metric with the general form ds 2 = dt 2 dx 2 , we have a b = diag (1, 1, 1, 1). The basis vectors of a nonholonomic basis satisfy ea eb = a b (5.4) Barcode Printer In Visual Basic .NET Using Barcode generation for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. GS1128 Printer In ObjectiveC Using Barcode printer for iPhone Control to generate, create GS1128 image in iPhone applications. In a nutshell, the basic idea of creating a nonholonomic basis is to scale it by the coef cient multiplying each differential in the line element. Let s illustrate this with an example. In the case of spherical polar coordinates, a noncoordinate basis is given by the following: er = r , 1 e = , r e = 1 r sin Reading Barcode In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Code 3 Of 9 Generator In Java Using Barcode creation for Java Control to generate, create ANSI/AIM Code 39 image in Java applications. An easy way to determine whether or not a given basis is holonomic is to calculate the commutation coef cients for the basis. We do this for the case of spherical polar coordinates in the next section. Generating ANSI/AIM Code 39 In VB.NET Using Barcode drawer for Visual Studio .NET Control to generate, create Code 39 Full ASCII image in .NET framework applications. Making Data Matrix ECC200 In None Using Barcode maker for Online Control to generate, create Data Matrix ECC200 image in Online applications. Commutation Coef cients
The commutator is de ned to be [A, B] = AB BA From calculus, we know that partial derivatives commute. Consider a function f (x, y). It is true that 2f 2f 2f 2f = =0 x y y x x y y x Cartan s Structure Equations
Let s rewrite this in terms of the commutator of the derivatives acting on the function f (x, y): 2f 2f = x y y x So we can write =0 , x y Looking at the holonomic basis for spherical polar coordinates, we had er = r = , r e = = , e = = x y y x f = , x y f From the above arguments, it is clear that [er , e ] = er , e = e , e = 0 Now let s consider the nonholonomic basis we found for spherical polar coordinates. Let s compute the commutator er , e . Since we re new to this process we carry along a test function as a crutch. 1 er , e f = r , r = = f = r 1 f r 1 ( r f ) r 1 1 1 f + ( r f ) ( r f ) 2 r r r 1 f r2
Using the de nitions given for the nonholonomic basis vectors, the end result is er , e f = 1 f r2 1 = e f r
Cartan s Structure Equations
The test function is just being carried along for the ride. Therefore, we can drop the test function and write 1 er , e = e r This example demonstrates that the commutators of a nonholonomic basis do not always vanish. We can formalize this in the following way: ei , e j = Ci j k ek (5.5) The Ci j k are called commutation coef cients. The commutation coef cients are antisymmetric in the rst two indices; i.e. Ci j k = C ji k If the following condition is met, then the basis set is holonomic. Ci j k = 0 i, j, k We can also compute the commutation coef cients using the basis one forms, as we describe in the next section.

