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qr code reader c# .net = sin cos = in .NET framework
= sin cos = QR Code Scanner In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. QR Code ISO/IEC18004 Creator In .NET Framework Using Barcode encoder for .NET framework Control to generate, create QR Code image in VS .NET applications. = cot
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Read Bar Code In .NET Framework Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications. Creating QR Code In C#.NET Using Barcode encoder for VS .NET Control to generate, create Quick Response Code image in .NET applications. Using the fact that the nonzero components of the Riemann tensor in two dizmensions are given by R1212 = R2121 = R1221 = R2112 , using (4.41) we calculate R = = = Since Paint QR Code ISO/IEC18004 In .NET Using Barcode printer for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Print Quick Response Code In VB.NET Using Barcode maker for VS .NET Control to generate, create QR Code image in .NET applications. + USS-128 Encoder In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create GS1 128 image in .NET framework applications. Bar Code Drawer In .NET Using Barcode creation for VS .NET Control to generate, create barcode image in VS .NET applications. a Making UPC - 13 In .NET Using Barcode generation for VS .NET Control to generate, create EAN-13 Supplement 5 image in .NET framework applications. Draw RM4SCC In .NET Using Barcode creation for Visual Studio .NET Control to generate, create RoyalMail4SCC image in .NET applications. a Painting Code128 In Java Using Barcode maker for Eclipse BIRT Control to generate, create Code 128 Code Set C image in BIRT reports applications. UPC Code Printer In None Using Barcode creation for Online Control to generate, create UPC-A Supplement 2 image in Online applications. = 0, this simpli es to R =
UPC-A Creator In None Using Barcode maker for Office Word Control to generate, create Universal Product Code version A image in Microsoft Word applications. ANSI/AIM Code 128 Generation In None Using Barcode creator for Online Control to generate, create Code 128B image in Online applications. = ( sin cos ) (cot ) ( sin cos ) = sin2 cos2 + cos sin (sin cos ) Scanning Bar Code In VB.NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications. GS1 128 Creation In VS .NET Using Barcode creation for ASP.NET Control to generate, create EAN128 image in ASP.NET applications. = sin2 cos2 + cos2 = sin2 The other nonzero components can be found using the symmetry R1212 = R2121 = R1221 = R2112 . Code-39 Generation In Java Using Barcode creation for Android Control to generate, create Code 39 Full ASCII image in Android applications. Generate EAN13 In Java Using Barcode generator for Java Control to generate, create EAN / UCC - 13 image in Java applications. The Ricci Tensor and Ricci Scalar
The Riemann tensor can be used to derive two more quantities that are used to de ne the Einstein tensor. The rst of these is the Ricci tensor, which is calculated from the Riemann tensor by contraction on the rst and third indices: Rab = R c acb (4.46) The Ricci tensor is symmetric, so Rab = Rba . Using contraction on the Ricci tensor, we obtain the Ricci scalar R = g ab Rab = R a a (4.47) Tensor Calculus
Finally, the Einstein tensor is given by G ab = Rab 1 Rgab 2
(4.48) EXAMPLE 4-12 Show that the Ricci scalar R = 2 for the unit 2-sphere. SOLUTION 4-12 In the previous example, we found that R = sin2 . The symmetry conditions of the Riemann tensor tell us that R = R = R = R The components of the metric tensor are g = g = 1, g = sin2 , g = 1 sin2 Applying the symmetry conditions, we nd R = sin2 R = sin2 R = sin2 Now we need to raise indices with the metric. This gives R = g R = 1 sin2 1 sin2 sin2 = 1 R = g R = sin2 R = g R = sin2 = 1
The components of the Ricci tensor are given by R = R c c = R + R = 1
Tensor Calculus
R = R c c = R + R = sin2 R = R = R c c = R + R = 0 Now we contract indices to get the Ricci scalar R = g ab Rab = g R + g R = 1 + 1 sin2 sin2 = 1 + 1 = 2 The Weyl Tensor and Conformal Metrics
We brie y mention one more quantity that will turn out to be useful in later studies. This is the Weyl tensor that can be calculated using the formula (in four dimensions) Cabcd = Rabcd + + 1 (gad Rcb + gbc Rda gac Rdb gbd Rca ) 2 (4.49) 1 (gac gdb gad gcb ) R 6
This tensor is sometimes known as the conformal tensor. Two metrics are conformally related if gab = 2 (x) gab (4.50) for some differentiable function (x). A metric is conformally at if we can nd a function (x) such that the metric is conformally related to the Minkowski metric gab = 2 (x) ab (4.51) A nice property of the Weyl tensor is that C a bcd is the same for a given metric and any metric that is conformally related to it. This is the origin of the term conformal tensor. Tensor Calculus
Quiz
For Questions 1 6, consider the following line element: ds 2 = dr 2 + r 2 d 2 + r 2 sin2 d 2 1. The components of the metric tensor are (a) grr = r, g = r sin , g = r 2 sin2 (b) grr = r, g = r 2 , g = r 2 sin2 (c) grr = 1, g = r 2 , g = r 2 sin2 (d) grr = r, g = r 2 , g = r 2 sin2 Compute the Christoffel symbols of the rst kind. (a) r 2 sin cos (b) r sin cos (c) r 2 sin2 (d) sin cos Now calculate the Christoffel symbols of the second kind. 1 (a) r 1 (b) r cos sin (c) cot (d) r12 Calculate the Riemann tensor. Rr is (a) sin (b) r 3 sin (c) rcos sin (d) 0 The determinant of the metric, g, is given by (a) r 2 sin4 (b) r 4 sin2 (c) r 4 sin4 (d) 0 Now let w a = (r, sin , sin cos ) and v a = r, r 2 cos , sin . 6. The Lie derivative u = L v w has u given by (a) r 2 cos2 cos sin (b) r cos2 cos sin
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