Essential Concepts

• In the cylindrical coordinate system, a point in space is represented by the ordered triple [latex](r,\theta,z)[/latex], where [latex](r,\theta)[/latex] represents the polar coordinates of the point’s projection in the [latex]xy[/latex]-plane and [latex]z[/latex] represents the point’s projection onto the [latex]z[/latex]-axis.
• To convert a point from cylindrical coordinates to Cartesian coordinates, use equations [latex]x=r\cos\theta[/latex], [latex]y=r\sin\theta[/latex], and [latex]z=z[/latex].
• To convert a point from Cartesian coordinates to cylindrical coordinates, use equations [latex]r^2=x^2+y^2[/latex], [latex]\tan\theta=\frac{y}{x}[/latex], and [latex]z=z[/latex].
• In the spherical coordinate system, a point [latex]P[/latex] in space is represented by the ordered triple [latex](\rho,\theta,\varphi)[/latex], where [latex]\rho[/latex] is the distance between [latex]P[/latex] and the origin [latex](\rho\ne{0})[/latex], [latex]\theta[/latex] is the same angle used to describe the location in cylindrical coordinates, and [latex]\varphi[/latex] is the angle formed by the positive [latex]z[/latex]-axis and line segment [latex]\overline{OP}[/latex], where [latex]O[/latex] is the origin and [latex]0\le\varphi\le\pi[/latex].
• To convert a point from spherical coordinates to Cartesian coordinates, use equations [latex]x=\rho\sin\varphi\cos\theta[/latex], [latex]y=\rho\sin\varphi\sin\theta[/latex], and [latex]z=\rho\cos\varphi[/latex].
• To convert a point from Cartesian coordinates to spherical coordinates, use equations [latex]\rho^{2}=x^2+y^2+z^2[/latex], [latex]\tan\theta=\frac{y}{x}[/latex], and [latex]\varphi=\arccos\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)[/latex]
• To convert a point from spherical coordinates to cylindrical coordinates, use equations [latex]r=\rho\sin\varphi[/latex], [latex]\theta=\theta[/latex], and [latex]z=\rho\cos\varphi[/latex].
• To convert a point from cylindrical coordinates to spherical coordinates, use equations [latex]\rho=\sqrt{r^2+z^2}[/latex], [latex]\theta=\theta[/latex], and [latex]\varphi=\arccos\left(\frac{z}{\sqrt{r^2+z^2}}\right)[/latex]