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polar and azimuthal angles

skatenerd

Active member
Oct 3, 2012
114
Kind of just a general question...I took a theoretical physics class last semester and we went through this whole book "Div, Grad, Curl, and All That Vector Calculus". Now I'm in an upper division E&M class and we're using Griffith's "Intro to Electrodynamics". In my 3rd semester of calculus and in the theoretical physics class, I got very accustomed to using phi as the angle coming from the z-axis to the xy-plane, and theta as the angle in the xy-plane.

Well I just got like half the problems wrong on our first homework to use spherical coordinates in E&M, because I didn't realize the author does it the other way around. While I'm a little frustrated, I'm mostly just looking to see what you all think is the most accepted version.

Is it more common/valid to use theta as the angle in the xy-plane or as the angle from the xy-plane to the z-axis? I feel like it makes more sense to me to have theta as the angle in the xy-plane because when I first learned polar coordinates in 2 dimensions, it was r and theta, and we were always working in the xy-plane.
Anyways, let me know what you guys think.
 

ThePerfectHacker

Well-known member
Jan 26, 2012
236
Is it more common/valid to use theta as the angle in the xy-plane or as the angle from the xy-plane to the z-axis? I feel like it makes more sense to me to have theta as the angle in the xy-plane because when I first learned polar coordinates in 2 dimensions, it was r and theta, and we were always working in the xy-plane.
Anyways, let me know what you guys think.
In USA, Calculus books are standardized and are all copy-and-pastes of one another (not that the books are any bad, the books have amazing quality to them, rather that all of them are essentially identical) with a few exceptions like Spivak and Apostol. In those Calculus books the standard notation for the $z$-axis angle is $\varphi$ and the standard notation for the polar angle is $\theta$.

I have a book, "Analytic Methods of Partial Differencial Equations", that uses $\varphi$ for the polar angle and $\theta$ for the $z$-axis angle. This book was written in the UK.

Perhaps, there is a different notational standard in the UK than in the USA.

Another common notational difference, inside the USA, is that the math courses all use $z$-axis as the axis going up while engineering courses all use $z$-axis as coming out of the board.
 

skatenerd

Active member
Oct 3, 2012
114
That's weird about the difference between engineering and maths x, y, and z axes. You would think since engineering tends to be more physically applied, they would want z to represent "up", right? Kind of odd...

I guess I should just try to get used to recognizing by reflex which angle is which no matter the convention the author is using. Thanks for the info
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
I've seen this issue a number of times before and seen various comments on it internationally.

It appears that the USA mathematics books are the odd ones out.
They define $\theta$ as the azimuthal angle with the x-axis and $\varphi$ as the angle with the z-axis (the co-latitude angle).

Most of the rest of the world, including USA engineering books, use $\varphi$ as the azimuthal angle and $\theta$ as the angle with the z-axis.

It means that in practice you always have to get the definitions straight, depending on context, and then apply them consistently.
 
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Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
In those Calculus books the standard notation for the $z$-axis angle is $\varphi$ and the standard notation for the polar angle is $\theta$.
The polar angle is the angle between the radius vector and the $z$ axis, i.e., the direction to the North pole (Wikipedia).

The international standard ISO 80000-2:2009 "Mathematical signs and symbols to be used in the natural sciences and technology", chapter 16, as well as its predecessor ISO 31-11 (see the "Coordinate systems" section) posits that $\varphi$ should denote the azimuth angle (on the reference plane) and $\theta$ (in fact, it uses $\vartheta$) should denote the polar angle. The ISO 80000-2 standard apparently costs 140 Swiss francs, but it is possible to find a copy online.

In Russia, the standard GOST 54521-2011 seems to follow ISO 80000-2 and also uses $\theta$ for the polar angle and $\varphi$ for the azimuth angle. I was teaching analytic geometry last semester, and the two textbooks I used used $\theta$ to denote the latitude, or elevation, i.e., the angle between the radius vector and the reference plane. If I get to teach this course next year, I'll follow the standard instead.
 
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