Question A.6.3: Consider a vector A = Ax x + Ay y +Az z at a point (x, y, z)...
Consider a vector A = A_{x} \hat{x} + A_{y} \hat{y} +A_{z} \hat{z} at a point (x, y, z) in Cartesian coordinates. Determine its location and representation in spherical coordinates.
From Table A.3 the location of (x, y, z) is r_{s} = \sqrt{x^{2}+y^{2} +z^{2}}, θ = cos^{-1}(z/\sqrt{x^{2}+y^{2} +z^{2}}), and Φ = tan^{-1}(y/x) . The vector is
TABLE A.3 Coordinate transformations for spherical coordinates
| Cylindrical | Cartesian | Spherical |
| = sin(θ) \hat{r}_{c} +cos(θ)\hat{z} | = sin(θ) cos(Φ)\hat{x} + sin(θ) sin(Φ)\hat{y} +cos(θ) \hat{z} | \hat{r}_{s} |
| = cos(θ) \hat{r}_{c} – sin(θ)\hat{z} | = cos(θ) cos(Φ)\hat{x} + cos(θ) sin(Φ)\hat{y} – sin(θ) \hat{z} | \hat{θ} |
| = \hat{Φ} | = – sin(Φ)\hat{x} + cos(Φ)\hat{y} | \hat{Φ} |
| = \sqrt{r_{c}^{2} +z^{2}} | = \sqrt{x^{2}+y^{2} +z^{2}} | r_{s} |
| = cos^{-1} (\frac{z}{\sqrt{r_{c}^{2} +z^{2}}}) | = cos^{-1} (\frac{z}{\sqrt{x^{2}+y^{2} +z^{2}}}) | θ |
| = Φ | = tan^{-1}(y/x) | Φ |
| = A_{r_{c}}sin(θ) +A_{z}cos(θ) | = A_{x}sin(θ)cos(Φ)+ A_{y}sin(θ)sin(Φ) +A_{z}cos(θ) | A_{r_{s}} |
| = A_{r}cos(θ) – A_{z}sin(θ) | = A_{x}cos(θ)cos(Φ)+ A_{y}cos(θ)sin(Φ) – A_{z}sin(θ) | A_{θ} |
| = A_{Φ} | = – A_{x}sin(Φ) +A_{y}cos(Φ) | A_{Φ} |
A = \underbrace{(A_{x}sin(θ)cos(Φ)+ A_{y}sin(θ)sin(Φ) +A_{z}cos(θ))}_{A_{r_{s}}}\hat{r}_{s}
+ \underbrace{(A_{x}cos(θ)cos(Φ)+ A_{y}cos(θ)sin(Φ) – A_{z}sin(θ))}_{A_{θ}} \hat{θ}
+ \underbrace{(-A_{x}sin(Φ) +A_{y}cos(Φ))}_{A_{Φ}} \hat{Φ}.
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