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qr code reader c# .net Fig. 24. An admittedly crude representation.The blob represents a manifold (basically a in .NET
Fig. 24. An admittedly crude representation.The blob represents a manifold (basically a Quick Response Code Reader In VS .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET framework applications. QR Code ISO/IEC18004 Generation In VS .NET Using Barcode generation for VS .NET Control to generate, create QR Code image in Visual Studio .NET applications. space of points). Tp is the tangent space at a point p. The tangent vectors live here.
Read QR In .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Drawing Barcode In Visual Studio .NET Using Barcode creator for .NET Control to generate, create bar code image in .NET applications. Vectors, One Forms, Metric
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UCC  12 Creation In None Using Barcode drawer for Office Word Control to generate, create UPC Code image in Office Word applications. Code 128C Maker In Java Using Barcode generation for Java Control to generate, create Code 128 image in Java applications. (2.4) We can do this because, as you know, it is possible to expand any vector in terms of some basis. What this relation gives us is an expansion of the basis vector ea in terms of the old basis eb . The components of ea in the eb basis are given by a b . Note that we are denoting the new coordinates by primes and the old coordinates are unprimed indices. EXAMPLE 21 Plane polar coordinates are related to cartesian coordinates by x = r cos and y = r sin Describe the transformation matrix that maps cartesian coordinates to polar coordinates, and write down the polar coordinate basis vectors in terms of the basis vectors of cartesian coordinates. Vectors, One Forms, Metric
SOLUTION 21a x Using a b = x b , the components of the transformation matrix are
x x r
x = cos r x = = r sin
and and
y = sin r y = = r cos
Using (2.4), we can write down the basis vectors in polar coordinates. We obtain er = e =
+ x ex +
r ex
r e y = cos ex + sin e y y e y = r sin ex + r cos e y
The components of a vector transform in the opposite manner to that of a basis vector (this is why, an ordinary vector is sometimes called contravariant; it transforms contrary to the basis vectors). This isn t so surprising given the placement of the indices. In particular, Va = a b bV =
xa b V xb
(2.5) The components of a one form transform as a = Basis one forms transform as a = dx a =
a b dx b b a
(2.6) (2.7) To nd the way an arbitrary tensor transforms, you just use the basic rules for vectors and one forms to transform each index (OK we aren t transforming the indices, but you get the drift). Basically, you add an appropriate for each index. For example, the metric tensor, which we cover in the next section, transforms as ga b = c a d b
The Metric
At the most fundamental level, one could say that geometry is described by the pythagorean theorem, which gives the distance between two points (see Vectors, One Forms, Metric
Fig. 25. The pythagorean theorem tells us that the lengths of a, b, c are related by
a2 + b 2 .
Fig. 25). If we call P1 = (x1 , y1 ) and P2 = (x2 , y2 ), then the distance d is given by d= (x1 x2 )2 + (y1 y2 )2 Graphically, of course, the pythagorean theorem gives the length of one side of a triangle in terms of the other two sides, as shown in Fig. 23. As we have seen, this notion can be readily generalized to the at spacetime of special relativity, where we must consider differences between spacetime events. If we label two events by (t1 , x1 , y1 , z 1 ) and (t2 , x2 , y2 , z 2 ), then we de ne the spacetime interval between the two events to be ( s)2 = (t1 t2 )2 (x1 x2 )2 (y1 y2 )2 (z 1 z 2 )2 Now imagine that the distance between the two events is in nitesimal. That is, if the rst event is simply given by the coordinates (t, x, y, z), then the second event is given by (t + dt, x + dx, y + dy, z + dz). In this case, it is clear that the differences between each term will give us (dt, dx, dy, dz). We write this in nitesimal interval as ds 2 = dt 2 dx 2 dy 2 dz 2 As we shall see, the form that the spacetime interval takes, which describes the geometry, is closely related to the gravitational eld. Therefore it s going to become quite important to familiarize ourselves with the metric. In short, the interval ds 2 , which often goes by the name the metric, contains information about how the given space (or spacetime) deviates from a at space (or spacetime). You are already somewhat familiar with the notion of a metric if you ve studied calculus. In that kind of context, the quantity ds 2 is often called a line element. Let s quickly review some familiar line elements. The most familiar is

