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qr code reader c# .net Tensor Operations in .NET
Tensor Operations Scanning QR Code In VS .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET applications. Encoding Denso QR Bar Code In VS .NET Using Barcode encoder for .NET Control to generate, create QR image in VS .NET applications. We now summarize a few basic algebraic operations that can be carried out with tensors to produce new tensors. These operations basically mirror the types of things you can carry out with vectors. For example, we can add two tensors of the same type to get a new tensor: R ab c = S ab c + T ab c Scanning QRCode In Visual Studio .NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Generating Barcode In .NET Framework Using Barcode encoder for .NET Control to generate, create bar code image in Visual Studio .NET applications. More on Tensors
Barcode Recognizer In .NET Framework Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications. Draw QR In C#.NET Using Barcode encoder for .NET framework Control to generate, create QR Code 2d barcode image in .NET framework applications. It follows that we can subtract two tensors of the same type to get a new tensor of the same type: Qa b = Sa b Ta b a Draw QR Code ISO/IEC18004 In VS .NET Using Barcode maker for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications. Painting QR Code JIS X 0510 In VB.NET Using Barcode encoder for VS .NET Control to generate, create QR Code image in .NET framework applications. We can also multiply a tensor by a scalar a to get a new tensor Sab = aTab Note that in these examples, the placement of indices and number of indices are arbitrary. We are simply providing speci c examples. The only requirement is that all of the tensors in these types of operations have to be of the same type. We can use addition, subtraction, and scalar multiplication to derive the symmetric and antisymmetric parts of a tensor. A tensor is symmetric if Bab = Bba and antisymmetric if Tab = Tba . The symmetric part of a tensor is given by T(ab) = 1 (Tab + Tba ) 2 (3.4) Generating 2D Barcode In VS .NET Using Barcode printer for .NET framework Control to generate, create Matrix 2D Barcode image in .NET applications. Code 3/9 Creation In Visual Studio .NET Using Barcode generation for .NET Control to generate, create Code 3 of 9 image in Visual Studio .NET applications. and the antisymmetric part of a tensor is T[ab] = 1 (Tab Tba ) 2 (3.5) Bar Code Creation In Visual Studio .NET Using Barcode creation for .NET framework Control to generate, create barcode image in .NET applications. Paint ABC Codabar In Visual Studio .NET Using Barcode drawer for .NET framework Control to generate, create ANSI/AIM Codabar image in .NET framework applications. We can extend this to more indices, but we won t worry about that for the time being. Often, the notation is extended to include multiple tensors. For instance, V(a Wb) = 1 (Va Wb + Wb Va ) 2 Draw European Article Number 13 In Java Using Barcode drawer for Java Control to generate, create European Article Number 13 image in Java applications. UCC.EAN  128 Generation In None Using Barcode encoder for Software Control to generate, create UCC  12 image in Software applications. Tensors of different types can be multiplied together. If we multiply a tensor of type (m, n) by a tensor of type ( p, q), the result is a tensor of type (m + p, n + q). For example R ab S c de = T abc de Contraction can be used to turn an (m, n) tensor into an (m 1, n 1) tensor. This is done by setting a raised and lowered index equal: Rab = R c acb Remember, repeated indices indicate a sum. (3.6) Encode UPC Symbol In None Using Barcode drawer for Office Excel Control to generate, create UPCA image in Excel applications. Barcode Decoder In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. More on Tensors
Paint UPCA Supplement 5 In VS .NET Using Barcode maker for ASP.NET Control to generate, create GTIN  12 image in ASP.NET applications. Create UCC  12 In None Using Barcode creator for Office Word Control to generate, create GS1 128 image in Microsoft Word applications. The Kronecker delta can be used to manipulate tensor expressions. Use the following rule: When a raised index in a tensor matches the lowered index in the Kronecker delta, change it to the value of the raised index of the Kronecker delta. This sounds confusing, so we demonstrate it with an example Data Matrix Generator In ObjectiveC Using Barcode creation for iPhone Control to generate, create Data Matrix 2d barcode image in iPhone applications. Barcode Decoder In Visual Basic .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET framework applications. a b T bc d = T ac d
Now consider the opposite. When a lowered index in a tensor matches a raised index in the Kronecker delta, set that index to the value of the lowered index of the Kronkecker delta: c d T ab c = T ab d
EXAMPLE 35 Show that if a tensor is symmetric then it is independent of basis. SOLUTION 35 We can work this out easily using the tensor transformation properties. Considering Bab = Bba , we work out the left side: Bab = c d b Bc d =
xc xd Bc d xa xb
For the other side, since we can move the derivatives around, we nd Bba =
d c a Bd c =
xd xc xc xd Bd c = Bd c xb xa xa xb
Equating both terms, it immediately follows that Bc d = Bd c . It is also true that if Bab = Bba , then B cd = B dc . Working this out, B cd = g ca B a d = g ca g db Bab = g ca g db Bba = g ca B d a = B dc EXAMPLE 36 Let T ab be antisymmetric. Show that S[a Tbc] = 1 (Sa Tbc Sb Tac + Sc Tab ) 3 More on Tensors
SOLUTION 36 Since T ab is antisymmetric, we know that Tab = Tba . The expression A[abc] is given by A[abc] = Therefore, we nd S[a Tbc] = 1 (Sa Tbc + Sb Tca + Sc Tab Sb Tac Sa Tcb Sc Tba ) 6 1 (Aabc + Abca + Acab Abac Aacb Acba ) 6 Now we use the antisymmetry of T , Tab = Tba , to write S[a Tbc] = 1 (Sa Tbc Sb Tac + Sc Tab Sb Tac + Sa Tbc + Sc Tab ) 6 1 = (2Sa Tbc 2Sb Tac + 2Sc Tab ) 6 1 = (Sa Tbc Sb Tac + Sc Tab ) 3 EXAMPLE 37 Let Q ab = Q ba be a symmetric tensor and R ab = R ba be an antisymmetric tensor. Show that Q ab Rab = 0 SOLUTION 37 Since R ab = R ba , we can write Rab = Therefore, we have Q ab Rab = 1 1 ab Q (Rab Rba ) = Q ab Rab Q ab Rba 2 2 1 1 (Rab + Rab ) = (Rab Rba ) 2 2 Note that the indices a, b are repeated in both terms. This means they are dummy indices and we are free to change them. In the second term, we make the switch

