How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?

How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?

Question
Alternate coordinate systems
asked 2020-12-28
How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?

Answers (1)

2020-12-29

Step 1 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 \(\displaystyle{\left({r},\theta,{z}\right)}\in{t}\hat{{r}}{\quad\text{and}\quad} \theta\) 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 (x, y, z) 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 \(\displaystyle{\left({p},\phi,\theta\right)}\) in which, p is the distance from P to the origin \((p \geq 0)\) \(\displaystyle\phi\) is 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 ctlindrical coordinates are, \(\displaystyle{r}={p}{\sin{\phi}}\)
\(\displaystyle{x}={r}{\cos{\theta}}={p}{\sin{\phi}}{\cos{\theta}}\)
\(\displaystyle{z}={p}{\cos{\phi}}\)
\(\displaystyle{y}={r}{\sin{\theta}}={p}{\sin{\phi}}{\sin{\theta}}\)
\(\displaystyle{p}=\sqrt{{{x}^{{{2}}}+{y}^{{{2}}}+{z}^{{{2}}}}}=\sqrt{{{r}^{{{2}}}+{z}^{{{2}}}}}\)
\(\displaystyle{\tan{\theta}}={\frac{{{y}}}{{{x}}}}\) 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 constant coordinate value.

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