I'm pretty sure that interpreting the spherical vector coefficients as a magnitude and two angles is not consistent with the adopted convention.

Suppose you had a vector (the one I'm working on at the moment)

\(\bf A=0.535 \hat{\bf x}-2.54 \hat{\bf z}\)

How would one express the equivalent vector in spherical polar coordinates at a point P(1,3,5)?

One approach would be to form a conversion matrix

\( \left[ \bf M \right ] =\left [ \begin{array}{cc} sin(\theta)cos(\phi) & sin(\theta)sin(\phi) & cos(\phi) \\ cos(\theta)cos(\phi) & cos(\theta)sin(\phi) & -sin(\theta) \\ -sin(\phi) & cos(\phi) & 0 \end{array}\right ] \)

Where

\(\theta \ and \ \phi\)

are part the spherical coordinates of P.

P(1,3,5)=>P(5.92,32.3°,71.6°)

One would substitute

\(\theta=32.3^o=0.564 rad\) and \( \phi=71.6^o=1.25 rad\)

to evaluate the terms for matrix M

To obtain

\( \left[ \bf M \right ] =\left [ \begin{array}{cc} 0.169 & 0.507 & 0.845 \\ 0.267 & 0.802 & -0.535 \\ -0.949 & 0.316 & 0 \end{array}\right ] \)

The spherical polar form coefficients of A at P would then be

\(\bf M \left [ 0.535 \\ 0 \\ -2.54 \right ]=\left [ -2.06 \\ 1.5 \\ -0.507\right ]\)

Giving A in spherical form as

\(\bf A=-2.06\hat{\bf \rho}+1.5\hat{\bf \theta}-0.507\hat{\bf \phi}\)

As far as I know this is a standard approach. Arguably the most commonly adopted one.

In reversing the process one wouldn't therefore interpret the 2nd and third coefficients of the spherical polar form of vector A as angles. All coefficients are rectilinear displacements.

An important cross-check would be to evaluate that the vector magnitude value is consistent between the two forms - Cartesian & Spherical

Suppose you had a vector (the one I'm working on at the moment)

\(\bf A=0.535 \hat{\bf x}-2.54 \hat{\bf z}\)

How would one express the equivalent vector in spherical polar coordinates at a point P(1,3,5)?

One approach would be to form a conversion matrix

\( \left[ \bf M \right ] =\left [ \begin{array}{cc} sin(\theta)cos(\phi) & sin(\theta)sin(\phi) & cos(\phi) \\ cos(\theta)cos(\phi) & cos(\theta)sin(\phi) & -sin(\theta) \\ -sin(\phi) & cos(\phi) & 0 \end{array}\right ] \)

Where

\(\theta \ and \ \phi\)

are part the spherical coordinates of P.

P(1,3,5)=>P(5.92,32.3°,71.6°)

One would substitute

\(\theta=32.3^o=0.564 rad\) and \( \phi=71.6^o=1.25 rad\)

to evaluate the terms for matrix M

\( \left[ \bf M \right ] =\left [ \begin{array}{cc} 0.169 & 0.507 & 0.845 \\ 0.267 & 0.802 & -0.535 \\ -0.949 & 0.316 & 0 \end{array}\right ] \)

The spherical polar form coefficients of A at P would then be

\(\bf M \left [ 0.535 \\ 0 \\ -2.54 \right ]=\left [ -2.06 \\ 1.5 \\ -0.507\right ]\)

Giving A in spherical form as

\(\bf A=-2.06\hat{\bf \rho}+1.5\hat{\bf \theta}-0.507\hat{\bf \phi}\)

As far as I know this is a standard approach. Arguably the most commonly adopted one.

In reversing the process one wouldn't therefore interpret the 2nd and third coefficients of the spherical polar form of vector A as angles. All coefficients are rectilinear displacements.

An important cross-check would be to evaluate that the vector magnitude value is consistent between the two forms - Cartesian & Spherical

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