# polar and azimuthal angles

#### skatenerd

##### Active member
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
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
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
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.

Last edited:

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
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.

Last edited: