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Unformatted text preview: 5 Vector calculus in spherical coordinates In studies of radiation from compact antennas it is more convenient to use spherical coordinates instead of the Cartesian coordinates that we are familiar with. In this lecture we will learn θ φ r cos θ r s i n θ r x y z r sin θ cos φ r sin θ sin φ 1. how to represent vectors and vector fields in spherical coordinates, 2. how to perform div, grad, curl, and Laplacian operations in spherical coordinates. • A 3D position vector r = ( x , y , z ) with Cartesian coordinates ( x , y , z ) is said to have spherical coordinates ( r , θ , φ ) where length r ≡ | r | = x 2 + y 2 + z 2 zenith angle θ = ta n- 1 x 2 + y 2 z azimuth angle φ = ta n- 1 y x . In terms of spherical coordinates, Cartesian coordi- nates can be expressed as x = r s in θ cos φ y = r s in θ s in φ z = r cos θ . Ratios x / r = s in θ cos φ , y / r = s in θ s in φ , and z / r = cos θ are referred to as direction cosines as they represent the cosine of the angle between vector r = ( x , y , z ) and the x-, y-, and z-axes, respectively. 1 • In Cartesian coordinates we have mutually orthogonal unit vectors ˆ x , ˆ y , ˆ z pointing in the direction of increasing Cartesian coordinates x , y , z , respectively. θ φ ˆ φ ˆ r ˆ θ r cos θ r s i n θ r x y z r sin θ cos φ r sin θ sin φ Unit-vectors ˆ r , ˆ θ , and ˆ φ shown in red, green, and blue point in mutually orthogonal direc- tions of increasing spherical coordinates r , θ , and φ , re- spectively, such that ˆ θ × ˆ φ = ˆ r . Note that ˆ r , ˆ θ , and ˆ φ are local unit vectors (i.e., coordinate dependent) unlike the global unit vectors ˆ x , ˆ y , and ˆ z of the Cartesian coordinate system....
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- Fall '08
- Electromagnet, Polar coordinate system, Coordinate systems, Vector field, Standard basis