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The function f is arbitrary, so we can write d dx a = d d x a in VS .NET
The function f is arbitrary, so we can write d dx a = d d x a Reading QR Code JIS X 0510 In VS .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET framework applications. Encode Quick Response Code In VS .NET Using Barcode maker for .NET framework Control to generate, create QR Code JIS X 0510 image in VS .NET applications. More on Tensors
Quick Response Code Recognizer In VS .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications. Bar Code Encoder In Visual Studio .NET Using Barcode generation for .NET Control to generate, create barcode image in .NET framework applications. d In other words, we have expanded the vector d in terms of a set of basis vectors. a x The components of the vector are dd and the basis vectors are given by a . x This is the origin of the argument that the basis vectors ea are given by partial derivatives along the coordinate directions. Barcode Decoder In .NET Framework Using Barcode reader for .NET Control to read, scan read, scan image in .NET framework applications. QR Code 2d Barcode Drawer In C#.NET Using Barcode printer for .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. EXAMPLE 34 The fact that the basis vectors are given in terms of partial derivatives with respect to the coordinates provides an explanation as to why we can write the transformation matrices as Encode Denso QR Bar Code In VS .NET Using Barcode creation for ASP.NET Control to generate, create QR Code ISO/IEC18004 image in ASP.NET applications. QR Generator In VB.NET Using Barcode generator for .NET framework Control to generate, create QRCode image in VS .NET applications. xa xb
Encoding Bar Code In Visual Studio .NET Using Barcode generation for .NET framework Control to generate, create barcode image in .NET framework applications. Encoding EAN128 In .NET Framework Using Barcode maker for .NET Control to generate, create EAN 128 image in .NET applications. SOLUTION 34 This can be done by applying the chain rule to a basis vector. We have, in the primed coordinates, ea = = a a x x xb xb = xb = xa xb Bar Code Generator In VS .NET Using Barcode generator for .NET Control to generate, create barcode image in Visual Studio .NET applications. Painting USS ITF 2/5 In VS .NET Using Barcode creator for .NET Control to generate, create I2/5 image in .NET framework applications. b a eb
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Code 3/9 Encoder In None Using Barcode maker for Software Control to generate, create ANSI/AIM Code 39 image in Software applications. Read European Article Number 13 In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Now let s explore some new notation that is frequently seen in books and the literature. We can write the dot product using a brackettype notation , . In the left slot, we place a one form and in the right side we place a vector. And so we write the dot product p v as p v = p, v = pa v a (3.3) Creating GS1  12 In None Using Barcode drawer for Microsoft Word Control to generate, create GS1  12 image in Word applications. Create Code39 In Java Using Barcode printer for Java Control to generate, create Code 3/9 image in Java applications. Here p is a one form. Using this notation, we can write the dot product between the basis one forms and basis vectors as a a , eb = b
More on Tensors
This can be seen easily writing the bases in terms of partial derivatives: a , eb = dx a , xa a = b = b xb x This type of notation makes it easy to nd the components of vectors and one forms. We consider the dot product between an arbitrary vector V and a basis one form: a a , V = a , V b eb = V b a , eb = V b b = V a
Since the components of a vector are just numbers, we are free to pull them outside of the bracket , . We can use the same method to nd the components of a one form: a , eb = a a , eb = a a , eb = a b = b
Now we see how we can derive the inner product between an arbitrary one form and vector: a , V = a a , V b eb = a V b a , eb = a V b b = a V a
These operations are linear. In particular, , aV + bW = a , V + b , W a + b , V = a , V + b , V where a, b are scalars, , are one forms, and V, W are vectors. Tensors as Functions
A tensor is a function that maps vectors and one forms to the real numbers. The components of a tensor are found by passing basis one forms and basis vectors as arguments. For example, we consider a rank 2 tensor T . If we pass two basis one forms as argument, we get T a , b = T ab More on Tensors
A tensor with raised indices has contravariant components and is therefore expanded in terms of basis vectors; i.e., T = T ab ea eb We have already seen an example of a tensor with lowered indices, the metric tensor gab a b = gab dx a dx b (in a coordinate basis). A tensor is not xed with raised or lowered indices; we recall from the last chapter that we can raise or lower indices using the metric. We can also have tensors with mixed indices. For each raised index we need a basis vector, and for each lowered index we need a basis one form when writing out the tensor as an expansion (the way you would write a vector expanding in a basis). For example S = S ab c ea eb c We get the components in the opposite way; that is, to get the upper index pass a one form, and to get the lower index pass a basis vector: S ab c = S a , b , ec We can pass arbitrary vectors and one forms to a tensor. Remember, the components of vectors and one forms are just numbers. So we can write S ( , , V ) = S a a , b b , V c ec = a b V c S a , b , ec = a b V c S ab c The quantity a b V c S ab c is a number, which is consistent with the notion that a tensor maps vectors and one forms to numbers. Note that the summation convention is used.

