![]() ![]() In vector calculus and electromagnetics work we often need to perform line, surface, and volume integrals. The base vectors meet the following relations: The base vector at P is per perpendicular to the plane of constant φ = φ 1.The base vector at P is perpendicular to the cone of constant θ = θ 1. ![]() The base vector at P is radial from the origin and is perpendicular to the sphere of constant R = R 1.A half-plane containing the z-axis and making an angle φ = φ 1 with the xz-plane (plane of constant φ).A right circular cone with its apex at the origin, its axis coinciding with the +z axis, and having a half-angle θ = θ 1 (cone of constant θ).A spherical surface centered at the origin with a radius R = R 1 (sphere of constant R).Ī point P(R 1, θ 1, φ 1) in spherical coordinates is located at the intersection of the following three surfaces: The base vector is perpendicular to the plane of constant z 1.The base vector at P is perpendicular to the half-plane surface of constant φ 1 and tangential to the cylindrical surface of constant r 1.The base vector at P is perpendicular to the cylindrical surface of constant r 1.A plane parallel to the xy -plane at z = z 1.A half-plane containing the z -axis and making angle φ = φ 1 with the xz-plane.In cylindrical coordinate systems a point P(r 1, θ 1, z 1) is the intersection of the following three surfaces as shown in the following figure. The base vectors meet the following relations: ,, and are the unit vectors in the three coordinate directions. A plane parallel to the x-y plane ( z = constant, normal to the z axis, unit vector ).A plane parallel to the x-z plane ( y = constant, normal to the y axis, unit vector ).A plane parallel to the y-z plane ( x = constant, normal to the x axis, unit vector ).In Cartesian coordinate system, a point is located by the intersection of the following three surfaces: If these three surfaces (in fact, their normal vectors) are mutually perpendicular to each other, we call them orthogonal coordinate system. U 1, u 2, and u 3 need not all be lengths as shown in the table below. ![]()
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