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RDCs provide orientational restraints rather than distance restraints, and has several advantages over NOEs: RDCs give information about the angle relative to the external magnetic field, which means that it can give information about the relative orientation of parts of the molecule that are far apart in the structure.
Sensors information (radar, sonar, and transponder data) is provided to the track algorithm using a polar coordinate system, and this is converted to cartesian coordinate system for the track algorithm. The polar to Cartesian conversion uses navigation data for sensors mounted on vehicles, which eliminates sensor position changes caused by ship ...
The radial coordinate is the most significant: geometrically it represents the geodesic distance to p of nearby points. Gauss's lemma asserts that the gradient of r is simply the partial derivative /. That is, , = for any smooth function ƒ. As a result, the metric in polar coordinates assumes a block diagonal form
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
Polar coordinates are used often in navigation as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is ...
Coordinates and ´ may be used to explain each other, because as we move along the coordinate line in one coordinate system we can use the other to describe our position. In this way Coordinates Z i {\displaystyle Z^{i}} and Z i ´ {\displaystyle Z^{\acute {i}}} are functions of each other
If this was just five years ago, let alone 10 or 20, the prospect of 72-year-old Bill Belichick as a college football coach would have been more about a splashy hire than the promise of great success.
Choosing a local coordinate system determines a local section of this bundle, which can then be used to pull back the Christoffel symbols to functions on M, though of course these functions then depend on the choice of local coordinate system. For each point, there exist coordinate systems in which the Christoffel symbols vanish at the point. [16]