Orthogonal Coordinate Systems - Cartesian, Cylindrical, and Spherical
Base Vectors
In a three-dimensional space, a point can be located as the intersection of three surfaces. The three surfaces are described by
u_{1}, u_{2}, and u_{3} need not all be lengths as shown in the table below.
u_{1} | u_{2} | u_{3} | |
Cartesian Coordinate System | x | y | z |
Cylindrical Coordinate System | r | φ | z |
Spherical Coordinate System | R | θ | φ |
If these three surfaces (in fact, their normal vectors) are mutually perpendicular to each other, we call them orthogonal coordinate system.
Cartesian Coordinate System:
In Cartesian coordinate system, a point is located by the intersection of the following three surfaces:
- A plane parallel to the y-z plane (x = constant, normal to the x 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 x-y plane (z = constant, normal to the z axis, unit vector )
This is shown in the figure below.
Base Vectors:
, , and are the unit vectors in the three coordinate directions. They are called the base vectors.
The base vectors meet the following relations:
Cylindrical Coordinate System:
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.
- A circular cylindrical surface r = r_{1}
- A half-plane containing the z-axis and making angle φ = φ_{1} with the xz-plane
- A plane parallel to the xy-plane at z = z_{1}.
Base Vectors:
- The base vector at P is perpendicular to the cylindrical surface of constant r_{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 is perpendicular to the plane of constant z_{1}.
The base vectors meet the following relations:
Spherical Coordinate System:
A point P(R_{1}, θ_{1}, φ_{1}) in spherical coordinates is located at the intersection of the following three surfaces:
- A spherical surface centered at the origin with a radius R = R_{1} (sphere of constant R)
- 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 half-plane containing the z-axis and making an angle φ = φ_{1} with the xz-plane (plane of constant φ)
This is shown below.
Base Vectors:
- The base vector at P is radial from the origin and is perpendicular to the sphere of constant R = R_{1}
- The base vector at P is perpendicular to the cone of constant θ = θ_{1}
- The base vector at P is per perpendicular to the plane of constant φ = φ_{1}
The base vectors meet the following relations:
Metric Coefficients
In vector calculus and electromagnetics work we often need to perform line, surface, and volume integrals. 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.
Line Integral
Surface Integral
Volume Integral
Cartesian 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 dθ, dφ.
Thus, we need a conversion factor to convert (mapping) a non-length based differential change (dθ, dφ, etc.) into a change in length dl as shown below.
dl_{i} = h_{i }du_{i}
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}.
u_{1}, u_{2}, u_{3} are coordinate axes in the chosen system, and correspond to x, y, z in Cartesian coordinate, r, φ, z in cylindrical coordinate, and R, θ, φ in spherical coordinate system.
Examples:
In the two-dimensional polar coordinates (u_{1}, u_{2}) = (r, φ), a differential change dφ (= du_{2}) corresponds to a differential length-change dl_{2} = r dφ (h_{2} = r = u_{1}).
In Cartesian coordinate systems, since dx, dy, dz are already length based, h_{1} = h_{2} = h_{3} = 1.
Differential Lengths
From the above discussion, we can see the differential length changes dl_{1}, dl_{2}, dl_{3 }are:
where h_{1}, h_{2}, h_{3} may be functions of u_{1}, u_{2}, and u_{3}.
A directed differential length-change dl in an arbitrary direction can be written as the vector sum of the component length-changes:
or
where a_{u1}, a_{u2}, a_{u3} are the base vectors.
Differential Area
In general orthogonal curvilinear coordinates the differential area ds_{1} (dA as shown in the figure above) normal to the unit vector a_{u1} is:
ds_{1} = dl_{2} dl_{3}
thus
ds_{1} = h_{2} h_{3} du_{2} du_{3}
Similarly, the differential areas normal to unit vectors a_{u2}, a_{u3} are:
ds_{2} = h_{1} h_{3} du_{1} du_{3}
ds_{3} = h_{1} h_{2} du_{1} du_{2}
For example, in Cartesian coordinate system:
ds_{x} = dy dz
ds_{y} = dz dx
ds_{z} = dx dy
where h_{1} = h_{2} = h_{3} = 1.
Differential Volume
The differential volume dv formed by differential coordinate changes du_{1}, du_{2}, and du_{3} in directions a_{u1}, a_{u2}, and a_{u3}, respectively, is (dl_{1} dl_{2} dl_{3}):
dv = dl_{1} dl_{2} dl_{3} = h_{1} h_{2} h_{3} du_{1} du_{2} du_{3}
Cartesian Coordinates
Cylindrical Coordinates
Spherical Coordinates
For convenience, the base vectors, metric coefficients, differential lengths, differential areas, and differential volume are listed in the following table.
Cartesian Coordinates (x, y, z) |
Cylindrical Coordinates (r, φ, z) |
Spherical Coordinates (R, θ, φ) |
||
Base Vectors | a_{u1}_{} | |||
a_{u2} | ||||
a_{u3}_{} | ||||
Metric Coefficients | h_{1} | 1 | 1 | 1 |
h_{2} | 1 | r | R | |
h_{3} | 1 | 1 | R sinθ | |
Differential Length | dl_{1} | dx | dr | dR |
dl_{2} | dy | r dφ | R dθ | |
dl_{3} | dz | dz | R sinθ dφ | |
Differential Area | ds_{1}_{} | dy dz | r dφ dz | R^{2} sinθ dθ dφ |
ds_{2}_{} | dx dz | dr dz | R sinθ dR dφ | |
ds_{3}_{} | dx dy | r dr dφ | R dR dθ | |
Differential Volume | dv | dx dy dz | r dr dφ dz | R^{2} sinθ dR dθ dφ |