- #1

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I have the following equations:

[tex]\left\{ \begin{array}{l}

x = \sin \theta \cos \varphi \\

y = \sin \theta \cos \varphi \\

z = \cos \theta

\end{array} \right.[/tex]

Assume [tex]\vec r = (x,y,z)[/tex], which is a 1*3 vector. Obviously, x, y, and z are related to each other. Now I want to calculate [tex]\frac{{\partial \vec r}}{{\partial z}}[/tex], could you please tell me if you have any hint?

I have Googled the questions a lot with different terms but can't find an answer that I am sure of. Many thanks for your attention!

Attempt 1: The problem seems related to coordinate transformation between spherical and cartesian coordinates.

Attempt 2: The problem seems related to "The Cartesian partial derivatives in spherical coordinates" shown here: http://mathworld.wolfram.com/SphericalCoordinates.html.

[tex]\left\{ \begin{array}{l}

x = \sin \theta \cos \varphi \\

y = \sin \theta \cos \varphi \\

z = \cos \theta

\end{array} \right.[/tex]

Assume [tex]\vec r = (x,y,z)[/tex], which is a 1*3 vector. Obviously, x, y, and z are related to each other. Now I want to calculate [tex]\frac{{\partial \vec r}}{{\partial z}}[/tex], could you please tell me if you have any hint?

I have Googled the questions a lot with different terms but can't find an answer that I am sure of. Many thanks for your attention!

Attempt 1: The problem seems related to coordinate transformation between spherical and cartesian coordinates.

Attempt 2: The problem seems related to "The Cartesian partial derivatives in spherical coordinates" shown here: http://mathworld.wolfram.com/SphericalCoordinates.html.

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