![]() Where i = (1, 2, or 3), h i is called the metric coefficient and may itself be a function of u 1, u 2, u 3. Thus, we need a conversion factor to convert (mapping) a non-length based differential change ( d θ, dφ, etc.) into a change in length dl as shown below. In these cases, we need to find the differential length change ( dl), differential area change ( ds), and differential volume change ( dv) in the chosen coordinate system.Ĭartesian coordinate system is length based, since dx, dy, dz are all lengths. However, in other curvilinear coordinate systems, such as cylindrical and spherical coordinate systems, some differential changes are not length based, such as d θ, dφ. 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. Sorry if it doesn't work out.U 1, u 2, and u 3 need not all be lengths as shown in the table below. I haven't checked this latter method, but I believe it agrees with the former one. Where $S(r)$ is the surface area of the sphere of radius $r$. Then we should get that our integral is equal to Each shell contributes the value of the function on that shell times the surface area of the shell. We may then express the integral as an integral in one variable - $r$ the radius of the shells (similar to how in a second semester of calculus one might calculate the volume of a surface of revolution with cylindrical shells). Perhaps a more enlightened approach to this problem would be to realize that the integrand is radially symmetric, so it is constant on spherical shells about the origin. We use the coordinate system $r, \phi_1.,\phi_\right)^n. Not a very clever approach, but we can compute this integral using hyper-spherical coordinates.
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