Question

BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es with a unit magnitude, in which case the differential displacement vector for the step we are considering would be ds =r do e, or (2) we can define a basis vector ds with magnitude r, so that we can write ds = d0 e. In each case, the magnitude of the displacement will be r do, but in the second case, the coordinate change de itself becomes the component of ds, which is convenient. This latter choice is the one that defines the “coordinate basis” vector for the direction in polar coordinates. The length of an infinitesimal step dr in the r direction (tangent to the 0 = constant curve) is simply dr, so if we define e, to have unit magnitude, we have ds = dr e, for such a step. Here, the basis vector with unit length is (in this case) the appropriate choice for a "coordinate basis” vector in the r direction. Once we have established these basis vectors, we can write the components of an arbitrary infinitesimal displacement in any direction as ds = dr e, + de er (5.18) Note carefully that this equation does not apply to finite displacements, but only displacements small enough so that the basis vectors e, and e, do not change sig- nificantly over the distance spanned by the displacement. (See the exercise below.) Note that by the nature of polar coordinates, basis vectors that point tangent to the 0 = constant and r= constant curves are perpendicular to each other at all points, but e, (for example) does not point in the same direction at one point as it does at another, as illustrated in figure 5.3. The metric for the polar coordinate basis is ere, e..e 101 8 uvex.e (5.19) ee, e.ee Important note: We can always specify the components of a metric tensor either by listing them in a matrix (as above) or by writing out the metric equation. For example, if we compare the abstract and concrete versions of the metric equation ds? = 8uv dx“ dx = dr? +r²d02 (5.20) we can immediately infer that gr = 1, 80o = r2, and gre = 80r =0 (because terms in- volving drd and do dr do not appear). The latter approach is often very convenient. ::-[ :)

Answer 1

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