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Fryderyk
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Homework Statement
As part of an assignment, I need to derive a transformation matrix to convert a vector in cartesian coordinates to spherical coordinates.
Homework Equations
What I've got so far is:
For an arbitrary vector V,
<br /> \textbf{V}=\left[\begin {array}{ccc}V_{x}&V_{y}&V_{z}\end{array}\right]\left[\begin{array}{c}\textbf{i}\\\textbf{j}\\\textbf{k}\end{array}\right] = \left[\begin{array}{ccc}V_{R}&V_{\theta}&V_{\phi}\end{array}\right]\left[\begin{array}{c}\textbf{e}_{R}\\\textbf{e}_{\theta}\\\textbf{e}_{\phi}\end{array}\right]<br />
Knowing that VR is the component of V in the eR direction, then:
<br /> V_{R}=aV_{x}+bV_{y}+cV_{z}<br />
where a, b and c are some transformation coefficients to be derived from geometry.
Similarly,
<br /> V_{\theta}=dV_{x}+eV_{y}+fV_{z}<br />
and
<br /> V_{\phi}=gV_{x}+hV_{y}+iV_{z}<br />
From this, a transformation matrix T exists such that:
<br /> \left[\begin{array}{ccc}V_{x}&V_{y}&V_{z}\end{array}\right]\textit{T}=\left[\begin{array}{ccc}aV_{x}+bV_{y}+cV_{z}&dV_{x}+eV_{y}+fV_{z}&gV_{x}+hV_{y}+iV_{z}\end{array}\right]=\left[\begin{array}{ccc}V_{R}&V_{\theta}&V_{\phi}\end{array}\right]<br />
The Attempt at a Solution
My problem is finding this matrix T.
It's obvious from inspection that:
<br /> T=\left[\begin{array}{ccc}a&d&g\\b&e&h\\c&f&i\end{array}\right]<br />
But, I'm not sure that just stating 'it's obvious that' will be sufficient.
I'm wondering if and how the below equation can be rearranged and solved to give T.
<br /> \left[\begin{array}{ccc}V_{x}&V_{y}&V_{z}\end{array}\right]\textit{T}=\left[\begin{array}{ccc}V_{R}&V_{\theta}&V_{\phi}\end{array}\right]<br />
After this point, finding the coefficients shouldn't be a problem.
Thank you
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