Below are various vectors in cartesian, cylindrical and spherical coordinates. Express the given vectors in two other coordinate systems outside the coordinate system in which they are expressed.

jernplate8 2021-08-02 Answered

Below are various vectors in cartesian, cylindrical and spherical coordinates. Express the given vectors in two other coordinate systems outside the coordinate system in which they are expressed
\(a) \overrightarrow{A}(x,y,z)=\overrightarrow{e}_{x}\)
\(d)\overrightarrow{A}(\rho, \phi, z)= \overrightarrow{e}_{\rho}\)
\(g)\overrightarrow{A}(r, \theta, \phi)=\overrightarrow{e}_{\theta}\)
\(j)\overrightarrow{A}(x,y,z)=\frac{-y\overrightarrow{e}_{x}+x\overrightarrow{e}_{y}}{x^{2}+y^{2}}\)

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Expert Answer

ottcomn
Answered 2021-08-03 Author has 22230 answers

Step 1
Given: \(a)\overrightarrow{A}(x,y,z)=\overrightarrow{e}_{x}\)
To express it in other two coordinate systems:
ie., in terms of cartesian, cylindrical and spherical coordinates
Step 2
Given: \(a)\overrightarrow{A}(x,y,z)=\overrightarrow{e}_{x}\)
spherical coordinates can be expressed as
\((r, \theta, \phi)=(\sqrt{x^{2}+y^{2}+z^{2}}, \tan^{-1} (\frac{y}{x}), \cos^{-1}(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}))\)
Therefore,
\(\displaystyle{\left({r},\theta,\phi\right)}={\left({1},{0},{\frac{{\pi}}{{{2}}}}\right)}\)

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