![]() Also, as with Cartesian unit vectors, any spherical unit vector dotted into itself gives unity, and crossed into itself gives zero. You should be able to prove this as well by using and from equation ( 1.3), crossing them, and seeing if emerges. Contrary to classical coordinates, such as Cartesian or. But a clockwise cross-product always gives a negative answer (try it with your right hand-your thumb will naturally point 'down'), so. This paper introduces a novel spherical (global) coordinate system that is free of singularity. Keep going clockwise until you hit the next unit vector in the triad, which is. Follow the circle until you get to by the shortest route this requires a clockwise motion. Suppose that you wish to compute Start by locating in the triad. Figure 1.2 shows pictorial triads as a way of remembering this. Like the Cartesian system, spherical coordinates form a right-handed system, with the same rules for forming scalar ('dot') and vector ('cross') products. Another important point is that unit vectors always point in the direction in which their corresponding coordinate increases see figure 1.1. Spherical coordinate unit vectors do not act like constants. This is important: Cartesian unit vectors are considered to be fixed in space and act like constants, while spherical coordinate unit vectors are functions of the direction (θ, ϕ) of the point under consideration.
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