qr code reader c# .net Vectors, One Forms, Metric in VS .NET

Printer QR in VS .NET Vectors, One Forms, Metric

the indices i, j will range only over (1, 2, 3). Typing a lot of Greek symbols is a bit of extra work, so we will stick to using Latin indices all the time. When possible, we will use early letters (a, b, c, . . .) to range over all possible values (0, 1, 2, 3) and use letters from the middle of the alphabet such as i, j to range only over the spatial components (1, 2, 3).

The Einstein summation convention is a way to write sums in a shorthand format. When the same index appears twice in an expression, once raised and once lowered, a sum is implied. As a speci c example,

An index that is summed over is called a dummy index, and can be replaced by another label if it is convenient. For example, S a Tab = S c Tcb The index b in the previous expressions is not involved in the sum operations. Such an index is known as a free index. A free index will typically appear on both sides of an expression. For example, consider the following equation: Aa =

The other index found in this expression, a , is a free index. If we elect to change the free index, it must be changed on both sides of the equation. Therefore it

would be valid to make the change a b , provided that we make this change on both sides; i.e., Ab =

We will often label basis vectors with the notation ea . Using the Einstein summation convention, a vector V can be written in terms of some basis as V = V a ea In this context the notation ea makes sense, because we can use it in the summation convention (this would not be possible with the cumbersome (i, j, k) for example). In a given coordinate system, the basis vectors ea are tangent to the coordinate lines. (See Fig. 2-3 and Fig. 2-4) This is the reason why we can write basis vectors as partial derivatives in a particular coordinate direction (for an explanation, see Carroll, 2004). In other words, we take ea = a = xa

This type of basis is called a coordinate basis. This allows us to think of a vector as an operator, one that maps a function into a new function that is related to its derivative. In particular, V f = (V a ea ) = V a a f A vector V can be represented with covariant components Va . This type of vector is called a one form. Basis one forms have raised indices and are often denoted by a . So we can write V = Va a We have used a tilde to note that this is a one form and not an ordinary vector (but it is the same object, in a different representation). Later, we will see how to move between the two representations by raising and lowering indices with the

metric. The basis one forms form a dual vector space to ordinary vectors, in the sense that the one forms constitute a vector space in their own right and the basis one forms map basis vectors to a number, the Kronecker delta function; i.e.,

(2.1)

In a coordinate representation, the basis one forms are given by a = dx a (2.2)

With this representation, it is easy to see why (2.1) holds. An arbitrary one form a maps a vector V a to a number via the scalar product V = a V a We can think of this either way: we can visualize vectors as maps that take one forms to the real numbers via the scalar product. More generally, we can de ne a ( p, q) tensor as a function that takes p one forms and q vectors as input and