<p>I have a doubt regarding a step in the derivation of the momentum equation in spherical coordinates. I am referring to the textbook "Fundamentals of Atmospheric Modeling" by Mark Jacobson, 2nd Edition, Chapter 4, Page 86. The change in unit vector in the x direction, delta i, is described using geometry. The text describes the angle between delta i and Re (The radius of Earth) as phi, the latitude. I do not understand how this can be so. Only after understanding this can I proceed to the expression for delta i in vector form. </p>

<p>Can anyone help please?<a href="https://i.sstatic.net/1cqQ9.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1cqQ9.png" alt="The coordinates"></a></p>

<p><a href="https://i.sstatic.net/6QMfn.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6QMfn.png" alt="The geometrical description"></a></p>

<p><a href="https://i.sstatic.net/KsQne.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/KsQne.png" alt="The equations, the one labelled 4.7 is what I do not understand"></a></p>

CC BY-SA 4.0
2020-05-06T21:31:05.327Z
$\Delta i_{\lambda}$ is defined, along with $i_{\lambda}$, to lie in a plane which is parallel towards the equatorial plane. The rest is fundamental trigonometry?
AtmosphericPrisonEscape
CC BY-SA 4.0
2020-05-07T02:50:02.800Z
Thank you. It sounds obvious but I was having a difficult time picturing it. But now it is better. All the three vectors, i, i + delta i and delta i, are all in a plane which is parallel to the equatorial plane. All the three vectors hence make the same angle phi with the Radius vector as is the definition of phi. And on resolving delta i into its components, the cos component is along Re and in turn along Re cos phi.
Anna Thomas
<atmosphere-modelling><coordinate-system>
Derivative of the Unit Vector in Spherical Coordinate system
Anna Thomas
# Derivative of the Unit Vector in Spherical Coordinate system

(Asked by: username_0 on 2020-05-06)

I have a doubt regarding a step in the derivation of the momentum equation in spherical coordinates. I am referring to the textbook "Fundamentals of Atmospheric Modeling" by Mark Jacobson, 2nd Edition, Chapter 4, Page 86. The change in unit vector in the x direction, delta i, is described using geometry. The text describes the angle between delta i and Re (The radius of Earth) as phi, the latitude. I do not understand how this can be so. Only after understanding this can I proceed to the expression for delta i in vector form.

Can anyone help please?[![The coordinates](https://i.sstatic.net/1cqQ9.png)](https://i.sstatic.net/1cqQ9.png)

[![The geometrical description](https://i.sstatic.net/6QMfn.png)](https://i.sstatic.net/6QMfn.png)

[![The equations, the one labelled 4.7 is what I do not understand](https://i.sstatic.net/KsQne.png)](https://i.sstatic.net/KsQne.png)

**Tags:** atmosphere-modelling, coordinate-system

> **Comments:**
> **username_1**: $\Delta i_{\lambda}$ is defined, along with $i_{\lambda}$, to lie in a plane which is parallel towards the equatorial plane. The rest is fundamental trigonometry?  
> **username_0**: Thank you. It sounds obvious but I was having a difficult time picturing it. But now it is better. All the three vectors, i, i + delta i and delta i, are all in a plane which is parallel to the equatorial plane. All the three vectors hence make the same angle phi with the Radius vector as is the definition of phi. And on resolving delta i into its components, the cos component is along Re and in turn along Re cos phi.  

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## 0 answers

