tinfoQ

2021-02-05

Provide notes on how triple integrals defined in cylindrical and spherical coordinates and the reason to prefer one of these coordinate systems to working in rectangular coordinates.

Gennenzip

Skilled2021-02-06Added 96 answers

Cylindrical coordinates prefer represent a point P in space by ordered triples $(r,\theta ,\text{}z)$ in which
r and $\theta$ are polar coordinates for the vertical projection of P on the $xy=pla\ne ,$ with $r\text{}\ge \text{}0$ , and
x is the rectangular vertical coordinate.

The equations relating rectangular$(x,\text{}y,\text{}z)$ and cylindrical $(r,\theta ,\text{}z)$ coordinates are,

$x=r\mathrm{cos}\theta$

$y=r\mathrm{sin}\theta$

$z=z$

$r}^{2}={x}^{2}\text{}+\text{}{y}^{2$

$\mathrm{tan}\theta =\frac{y}{x}$

The spherical coordinates represent a point P in by ordered triples$(\rho ,\varphi ,\theta )$ in which,

$\rho$ is the distance from P to the origin $(\rho \text{}\ge \text{}0)$

.$\varphi$ is the angle $over\to \left\{OP\right\}$ makes with the positive z - axis $(0\text{}\le \text{}\varphi \text{}\le \text{}\pi )$

.$\theta$ is the angle from cylindrical coordinates.

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

$r=\rho \mathrm{sin}\varphi$

$x=r\mathrm{cos}\theta =\rho \mathrm{sin}\varphi \mathrm{cos}\theta$

$z=\rho \mathrm{cos}\varphi$

$y=r\mathrm{sin}\theta =\rho \mathrm{sin}\varphi \mathrm{sin}\theta$

$\rho =\sqrt{{x}^{2}\text{}+\text{}{y}^{2}\text{}+\text{}{z}^{2}}=\sqrt{{r}^{2}\text{}+\text{}{z}^{2}}$

$\mathrm{tan}\theta =\frac{y}{x}$

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.

The equations relating rectangular

The spherical coordinates represent a point P in by ordered triples

.

.

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

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.

An object moving in the xy-plane is acted on by a conservative force described by the potential energy function

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