Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

7.2: Spherical Coordinates

( \newcommand{\kernel}{\mathrm{null}\,}\)

Spherical coordinates are defined from Cartesian coordinates as

r=x2+y2+z2ϕ=arctan(y/x)θ=arctan(x2+y2z)

or alternatively

x=rcosϕsinθ,y=rsinϕsinθz=rcosθ

as indicated schematically in Fig. 7.2.1.

spherical.png
Figure 7.2.1: Spherical coordinates

Using the chain rule we find

x =rxr +ϕxϕ +θxθ =xrr yx2+y2ϕ +xzr2x2+y2θ =sinθcosϕr sinϕrsinθϕ +cosϕcosθrθ ,y =ryr +ϕyϕ +θyθ =yrr +xx2+y2ϕ +yzr2x2+y2θ =sinθsinϕr +cosϕrsinθϕ +sinϕcosθrθ ,z =rzr +ϕzϕ +θzθ =zrr x2+y2r2θ =sinθsinϕr sinθrθ .

once again we can write in terms of these coordinates.

=ˆerr +ˆeϕ1rsinθϕ +ˆeθ1rθ  where the unit vectors ˆer=(sinθcosϕ,sinθsinϕ,cosθ),ˆeϕ=(sinϕ,cosϕ,0),ˆeθ=(cosϕcosθ,sinϕcosθ,sinθ).

are an orthonormal set. We say that spherical coordinates are orthogonal.

We can use this to evaluate Δ=2,

Δ=1r2r (r2r )+1r21sinθθ (sinθθ )+1r22ϕ2 

Finally, for integration over these variables we need to know the volume of the small cuboid contained between r and r+δr, θ and θ+δθ and ϕ and ϕ+δϕ.

spherical2.png
Figure 7.2.2: Integration in spherical coordinates

The length of the sides due to each of these changes is δr, rδθ and rsinθδθ, respectively (these are the Jacobians for the conversion of Cartesian coordinates to polar and spherical coordinates, respectively). We thus conclude that

Vf(x,y,z)dxdydz=Vf(r,θ,ϕ)r2sinθdrdθdϕ.

Contributors and Attributions


This page titled 7.2: Spherical Coordinates is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?