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Unformatted text preview: 5 Vector Calculus in spherical coordinates In radiation studies, spherical coordinates are more convenient to use than the Cartesian coordinates. In spherical coordinate system, the position of a point ) , , ( z y x r is given by three constant value surface: A sphere, a cone, a cylinder, and a plane given by angle azimuth tan angle zenith tan origin from distance 1 2 2 1 2 2 2 x y z y x z y x r r Cartesian coordinates in terms of spherical coordinates are cos sin sin cos sin r z r y r x Unit vectors are vectors that have unit length and directions determined by tangents to constant value surfaces at the point. Unit vectors in Cartesian coordinates are z y x ˆ , ˆ , ˆ . Unit vectors in Cartesian coordinates are . ˆ , ˆ , ˆ r These constitute a right-system with the following product rules: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ r r r ˆ ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ ˆ ˆ ˆ r r r r Conversion between spherical and Cartesian coordinates and vectors are very easy once we know the dot multiplications of the unit vectors in the two coordinate systems: cos ˆ ˆ sin sin ˆ ˆ cos sin ˆ ˆ z r y r x r sin ˆ ˆ sin cos ˆ ˆ cos cos ˆ ˆ z y x ˆ ˆ cos ˆ ˆ sin ˆ ˆ z y x...
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This note was uploaded on 08/25/2011 for the course ECE 350 taught by Professor Kudeki during the Spring '11 term at University of Illinois at Urbana–Champaign.
- Spring '11