Give the notes about how the triple integrals defined in cylindrical and spherical coordinates.

Step 2

The cylindrical coordinates denotes a point P in space by ordered triples \(\displaystyle{\left({r},\theta,{z}\right)}\) in that r and \(\displaystyle\theta\) are polar coordinates for the vertical projection of P on the xy-plane with \(\displaystyle{r}\geq\theta\) and z is the rectangular vertical coordinate.

The equations related to the rectangular coordinates \(\displaystyle{\left({x},{y},{z}\right)}\) and cylindrical coordinates \(\displaystyle{\left({r},\theta,{z}\right)}\) are,

\(\displaystyle{x}={r}{\cos{\theta}},{y}={r}{\sin{\theta}},{z}={z},{r}^{{{2}}}={x}^{{{2}}}+{y}^{{{2}}}\) and \(\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}\)

Step 3

The spherical coordinates represent a point P in space by ordered triples \(\displaystyle{\left(\rho,\phi,\theta\right)}\) in which,

\(\displaystyle\rho\) is the distance from P to the origin \(\displaystyle{\left(\rho\geq{0}\right)}\).

\(\displaystyle\phi\) is the angle \(\displaystyle{o}{v}{e}{r}\rightarrow{\left\lbrace{O}{P}\right\rbrace}\) makes with the positive z-axis \(\displaystyle{\left({0}\leq\phi\leq\pi\right)}\)

\(\displaystyle\theta\) is the angle from cylindrical coordinates.

Step 4

The equations relating spherical coordinates to Cartesian and cylindrical coordinates are,

\(\displaystyle{r}=\rho{\sin{\phi}}\)

\(\displaystyle{x}={r}{\cos{\theta}}=\rho{\sin{\phi}}{\cos{\theta}}\)

\(\displaystyle{z}=\rho{\cos{\phi}}\)

\(\displaystyle{r}{\sin{\theta}}=\rho{\sin{\phi}}{\sin{\theta}}\)

\(\displaystyle\rho=\sqrt{{{x}^{{{2}}}+{y}^{{{2}}}+{z}^{{{2}}}}}=\sqrt{{{r}^{{{2}}}+{z}^{{{2}}}}}\)

\(\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}\)

Cylindrical coordinates are good for describing cylinders whose axes run along the z-axis and planes that either contain the z-axis or lie perpendicular to the z-axis. Surfaces like these have equations of constant coordinate value.