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

u₁, u₂, and u₃ need not all be lengths as shown in the table below.

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(r₁, θ₁, z₁) is the intersection of the following three surfaces as shown in the following figure.

• A circular cylindrical surface r = r₁
• A half-plane containing the z-axis and making angle φ = φ₁ with the xz-plane
• A plane parallel to the xy-plane at z = z₁.

Base Vectors:

• The base vector at P is perpendicular to the cylindrical surface of constant r₁.
• The base vector   at P is perpendicular to the half-plane surface of constant φ₁ and tangential to the cylindrical surface of constant r₁.
• The base vector  is perpendicular to the plane of constant z₁. 

The base vectors meet the following relations:

Spherical Coordinate System:

A point P(R₁, θ₁, φ₁) in spherical coordinates is located at the intersection of the following three surfaces:

• A spherical surface centered at the origin with a radius R = R₁ (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 = θ₁ (cone of constant θ)
• A half-plane containing the z-axis and making an angle φ = φ₁ 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 = R₁
• The base vector at P is perpendicular to the cone of constant θ = θ₁
• The base vector  at P is per perpendicular to the plane of constant φ = φ₁ 

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  u₁, u₂, u₃.

u₁, u₂, u₃ 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 (u₁, u₂) = (r, φ), a differential change dφ (= du₂) corresponds to a differential length-change dl₂ = r dφ (h₂ = r = u₁).

In Cartesian coordinate systems, since dx, dy, dz are already length based, h₁ = h₂ = h₃ = 1.

Differential Lengths

From the above discussion, we can see the differential length changes dl₁, dl₂, dl₃ are:

where h₁, h₂, h₃ may be functions of u₁, u₂, and u₃.

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 ds₁ (dA as shown in the figure above) normal to the unit vector a_u1 is:

ds₁ = dl₂ dl₃

thus

ds₁ = h₂ h₃ du₂ du₃

Similarly, the differential areas normal to unit vectors a_u2, a_u3 are:

ds₂ = h₁ h₃ du₁ du₃

ds₃ = h₁ h₂ du₁ du₂ 

For example, in Cartesian coordinate system:

ds_x = dy dz

ds_y = dz dx

ds_z = dx dy

where h₁ = h₂ = h₃ = 1.

Differential Volume

The differential volume dv formed by differential coordinate changes du₁, du₂, and du₃ in directions a_u1, a_u2, and a_u3, respectively, is (dl₁ dl₂ dl₃):

dv = dl₁ dl₂ dl₃ = h₁ h₂ h₃ du₁ du₂ du₃

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	au1
	au2
	au3
Metric Coefficients	h1	1	1	1
	h2	1	r	R
	h3	1	1	R sinθ
Differential Length	dl1	dx	dr	dR
	dl2	dy	r dφ	R dθ
	dl3	dz	dz	R sinθ dφ
Differential Area	ds1	dy dz	r dφ dz	R2 sinθ dθ dφ
	ds2	dx dz	dr dz	R sinθ dR dφ
	ds3	dx dy	r dr dφ	R dR dθ
Differential Volume	dv	dx dy dz	r dr dφ dz	R2 sinθ dR dθ dφ