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In addition to the well‐known Cartesian coordinates, the spherical coordinates (r, θ, ϕ), as shown in Figure 1.10, will also be used frequently throughout this book. These two coordinate systems have the following relations:

(1.11)

and

(1.12)

Figure 1.10 Cartesian and spherical coordinates

The dot products of unit vectors in these two coordinator systems are

(1.13)

Thus, we can express a quantity in one coordinate system using the known parameters in the other coordinate system. For example, if A_r, A_θ, A_φ are known, we can find