91影视

The spherical coordinates are defined in terms of $r,胃,蠒$, where r is the distance from origin, $胃$is the pole angle and $蠒$ is the azimuthal angle.

The spherical coordinates can be drawn as,

The scalar potentials is $v=\frac{{\displaystyle \stackrel{鈭�}{r}}}{{\displaystyle r^2}}$ and the position vector is $\stackrel{鈫�}{r}=x鈥�\text{i}+y鈥�\text{j}+z鈥�\text{k}$. The unit vector in the direction of $\stackrel{鈫�}{r}$, is obtained as,

$\begin{array}{rcl}\stackrel{鈭�}{r}& =& \frac{\stackrel{鈫�}{r}}{\left|r\right|}\\ & =& \frac{x鈥�\text{i}+y鈥�\text{j}+z鈥�\text{k}}{\left|\sqrt{x^2+y^2+z^2}\right|}\end{array}$

The spherical coordinates of the system is defined as,

$x=\left(r\sin 胃\right)\cos 蠒$

$y=\left(r\sin 胃\right)\sin 蠒$

$z=r\cos 胃$

Substitute $\left(r\sin 胃\right)\cos 蠒$for x , $\left(r\sin 胃\right)\sin 蠒$for y and $r\cos 胃$ for z into

$\stackrel{鈫�}{r}=x鈥�\text{i}+y鈥�\text{j}+z鈥�\text{k}$

$\begin{array}{rcl}\stackrel{鈫�}{r}& =& x鈥�\text{i}+y鈥�\text{j}+z鈥�\text{k}\\ & =& \left(r\sin 胃\right)\cos 蠒鈥�\text{i}+\left(r\sin 胃\right)\sin 蠒鈥�\text{j}+r\cos 胃鈥�k\\ & & \end{array}$

The unit vector $\stackrel{鈭�}{r}$is obtained as$\stackrel{鈭�}{r}=\sin 胃\cos 蠒鈥�\stackrel{鈭�}{x}+\sin 胃\sin 蠒\stackrel{鈭�}{y}+\cos 胃\stackrel{鈭�}{z}$

The infinitesimal displacement along the direction$胃$, is obtained as ${\stackrel{鈫�}{dl}}_{胃}=rd胃\stackrel{鈭�}{胃}$ 鈥︹��. (3)

The infinitesimal displacement along the direction $胃$, in terms of Cartesian coordinates is written as,

${\stackrel{鈫�}{dl}}_{胃}=dx\stackrel{鈭�}{x}+dy\stackrel{鈭�}{y}+dz\stackrel{鈭�}{x}$

As $x=\left(r\sin 胃\right)\cos 蠒$ , $y=\left(r\sin 胃\right)\sin 蠒$, $z=r\cos 胃$, infinitesimal displacement along the direction $胃$, can be written as,

${\stackrel{鈫�}{dl}}_{胃}=\left(\left(r\sin 胃\right)\cos 蠒\right)\stackrel{鈭�}{x}+\left(\left(r\sin 胃\right)\sin 蠒\right)\stackrel{鈭�}{y}+\left(r\cos 胃\right)\stackrel{鈭�}{z}$

From equation (3), ${\stackrel{鈫�}{dl}}_{胃}=rd胃\stackrel{鈭�}{胃}$

$rd胃鈥�\stackrel{鈭�}{胃}=\left(\left(r\sin 胃\right)\cos 蠒\right)\stackrel{鈭�}{x}+\left(\left(r\sin 胃\right)\sin 蠒\right)\stackrel{鈭�}{y}+\left(r\cos 胃\right)\stackrel{鈭�}{z}$

The infinitesimal displacement along the direction $胃$, is obtained as

${\stackrel{鈫�}{dl}}_{蠒}=r\sin 胃d蠒\stackrel{鈭�}{蠒}$ 鈥︹��. (3)

The infinitesimal displacement along the direction $胃$, in terms of Cartesian coordinates is written as,

${\stackrel{鈫�}{dl}}_{胃}=dx鈥�\stackrel{鈭�}{x}+dy\stackrel{鈭�}{y}+dz\stackrel{鈭�}{z}$

As $x=\left(r\sin 胃\right)\cos 蠒$, $y=\left(r\sin 胃\right)\sin 蠒$, $z=r\cos 胃$, infinitesimal displacement along the direction $胃$, can be written as,

${\stackrel{鈫�}{dl}}_{胃}=\left(\left(r\sin 胃\right)\cos 蠒\right)\stackrel{鈭�}{x}+\left(\left(r\sin 胃\right)\sin 蠒\right)\stackrel{鈭�}{y}鈥�+\left(r\cos 胃\right)\stackrel{鈭�}{z}$

From equation (3), ${\stackrel{鈫�}{dl}}_{蠒}=r\sin 胃d蠒\stackrel{鈭�}{蠒}$

$\begin{array}{rcl}r\sin 胃d蠒\stackrel{鈭�}{蠒}& =& -\left(\left(r\sin 胃\right)\cos 蠒\right)\stackrel{鈭�}{x}+\left(\left(r\sin 胃\right)\sin 蠒\right)\stackrel{鈭�}{y}\\ \stackrel{鈭�}{蠒}& =& 鈥�鈥�-\sin 蠒\stackrel{鈭�}{x}+\cos 蠒鈥�\stackrel{鈭�}{y}\end{array}$

The product of $\stackrel{鈭�}{r}.\stackrel{鈭�}{r}$, is calculated as,

$\begin{array}{rcl}\stackrel{鈭�}{r}.\stackrel{鈭�}{r}& =& {\sin }^2胃\left({\cos }^2蠒+{\sin }^2蠒\right)+{\cos }^2胃\\ & =& {\sin }^2胃+{\cos }^2胃\\ & =& 1\\ & & \end{array}$

Multiply the vectors $\stackrel{鈭�}{胃}$ and $\stackrel{鈭�}{蠒}$

$\begin{array}{rcl}\stackrel{鈭�}{胃}.\stackrel{鈭�}{蠒}& =& -\cos 胃\sin 蠒\cos 蠒+\cos 胃\sin 蠒\cos 蠒\\ & =& 0\\ & & \end{array}$

As$x=\left(r\sin 胃\right)\cos 蠒$, $y=\left(r\sin 胃\right)\sin 蠒$, $z=r\cos 胃$, the position vector

$\stackrel{鈭�}{r}=\left(r\sin 胃\right)\cos 鈥�\stackrel{鈭�}{x}+\sin 胃\sin 蠒\stackrel{鈭�}{y}+\cos 胃\stackrel{鈭�}{z}$

Multiply above equation by $\sin 胃$on both sides,

$\sin 胃\stackrel{鈭�}{r}={\sin }^2胃\cos \stackrel{鈭�}{x}鈥�+{\sin }^2胃\sin 蠒\stackrel{鈭�}{y}+\sin 胃\cos 胃鈥�\stackrel{鈭�}{z}$ 鈥︹��. (1)

Now the theta vector is $\stackrel{鈭�}{胃}=\cos 胃\cos 蠒\stackrel{鈭�}{x}+\cos 胃\sin 蠒\stackrel{鈭�}{y}-\sin 胃\stackrel{鈭�}{z}$

Multiply above equation by $\cos 蠒$on both sides,

$\cos 胃\stackrel{鈭�}{胃}={\cos }^2胃\cos 蠒鈥�\stackrel{鈭�}{x}+{\cos }^2胃\sin 蠒\stackrel{鈭�}{y}-\sin 胃\cos 胃鈥�\stackrel{鈭�}{z}$ 鈥︹��. (2)

Add equations (1) and (2) as,

$\begin{array}{rcl}\sin 胃\stackrel{鈭�}{r}+\cos 胃\stackrel{鈭�}{胃}& =& {\sin }^2胃\cos 蠒\stackrel{鈭�}{x}+{\sin }^2胃\sin 蠒\stackrel{鈭�}{y}+\sin 胃\cos 胃\stackrel{鈭�}{z}+{\cos }^2胃\cos 蠒\stackrel{鈭�}{x}+{\cos }^2胃\sin 蠒\stackrel{鈭�}{y}-\sin 胃\cos 胃\stackrel{鈭�}{z}\\ & =& {\sin }^2胃\cos 蠒\stackrel{鈭�}{x}+{\sin }^2胃\sin 蠒\stackrel{鈭�}{y}+{\cos }^2胃\cos 蠒\stackrel{鈭�}{x}+{\cos }^2胃\sin 蠒\stackrel{鈭�}{y}\\ & =& \left({\sin }^2胃+{\cos }^2胃\stackrel{鈭�}{x}\right)\cos 蠒\stackrel{}{x+\left({\sin }^2胃+{\cos }^2胃\right)\sin 蠒\stackrel{鈭�}{y}}\\ & =& \cos 蠒\stackrel{鈭�}{x}+\sin 蠒\stackrel{鈭�}{y}\end{array}$

solve further as,

$\stackrel{鈭�}{蠒}=-\sin 蠒\stackrel{鈭�}{x}+\cos 蠒\stackrel{鈭�}{y}$

Multiply $\sin 胃\stackrel{鈭�}{r}+\cos 胃\stackrel{鈭�}{胃}=\cos 蠒\stackrel{鈭�}{x}+\sin 蠒\stackrel{鈭�}{y}$ by $\cos 蠒$on both sides,

$\sin 胃\cos 蠒\stackrel{鈭�}{r}+\cos 胃\cos 蠒\stackrel{鈭�}{胃}={\cos }^2蠒\stackrel{鈭�}{x}+\sin 蠒\cos 蠒\stackrel{鈭�}{y}$ 鈥︹��. (3)

Multiply $\stackrel{鈭�}{蠒}=-\sin 蠒\stackrel{鈭�}{x}+\cos 蠒\stackrel{鈭�}{y}$by $\sin 蠒$on both sides,

$\sin 蠒\stackrel{鈭�}{蠒}=-{\sin }^2蠒\stackrel{鈭�}{x}+\cos 蠒\sin 蠒\stackrel{鈭�}{y}$ 鈥︹��. (4)

Subtract equation (4) from equation (3).

$\begin{array}{rcl}\sin 胃\cos 蠒\stackrel{鈭�}{r}+\cos 胃\cos 蠒\stackrel{鈭�}{胃}-\sin 蠒\stackrel{鈭�}{蠒}& =& {\cos }^2蠒\stackrel{鈭�}{x}+\sin 蠒\cos 蠒\stackrel{鈭�}{y}-\sin 蠒\cos 蠒\stackrel{鈭�}{y}+{\sin }^2蠒\stackrel{鈭�}{x}\\ & =& {\cos }^2蠒\stackrel{鈭�}{x}+{\sin }^2蠒\stackrel{鈭�}{x}\\ & =& \stackrel{鈭�}{x}\end{array}$

Thus, $\stackrel{鈭�}{x}=\sin 胃\cos 蠒\stackrel{鈭�}{r}+\cos 胃\cos 蠒\stackrel{鈭�}{胃}-\sin 蠒\stackrel{鈭�}{蠒}$

Multiply $\sin 胃\stackrel{鈭�}{r}+\cos 胃\stackrel{鈭�}{胃}=\cos 蠒\stackrel{鈭�}{x}鈥�+\sin 蠒\stackrel{鈭�}{y}$by $\sin 蠒$ on both sides,

$\sin 胃\sin 蠒\stackrel{鈭�}{r}+\cos 胃\sin 蠒\stackrel{鈭�}{胃}=\cos 蠒\sin 蠒鈥�\stackrel{鈭�}{x}+{\sin }^2蠒\stackrel{鈭�}{y}$ 鈥︹��. (5)

Multiply $\stackrel{鈭�}{蠒}=-\sin 蠒鈥�\stackrel{鈭�}{x}+\cos 蠒\stackrel{鈭�}{y}$ by $\cos 蠒$ on both sides,

$\cos 蠒\stackrel{鈭�}{蠒}=-\sin 蠒鈥�\cos 蠒\stackrel{鈭�}{x}+{\cos }^2蠒\stackrel{鈭�}{y}$ 鈥︹��. (5)

Add equation (5) and (6).

$\begin{array}{rcl}\sin 胃\sin 蠒\stackrel{鈭�}{r}+\cos 胃\sin 蠒\stackrel{鈭�}{胃}+\cos 蠒\stackrel{鈭�}{蠒}& =& \cos 蠒\sin 蠒\stackrel{鈭�}{x}+{\sin }^2蠒\stackrel{鈭�}{y}-\sin 蠒鈥�\cos 蠒\stackrel{鈭�}{x}+{\cos }^2蠒\stackrel{鈭�}{y}\\ & =& {\sin }^2蠒\stackrel{鈭�}{y}+{\cos }^2蠒\stackrel{鈭�}{y}\\ & =& \stackrel{鈭�}{y}\end{array}$

Thus, $\stackrel{鈭�}{y}=\begin{array}{rcl}& & \sin \end{array}\begin{array}{rcl}& & 胃\end{array}\begin{array}{rcl}& & \sin \end{array}\begin{array}{rcl}& & 蠒\end{array}\begin{array}{rcl}& & \stackrel{鈭�}{r}\end{array}\begin{array}{rcl}& & +\end{array}\begin{array}{rcl}& & \cos \end{array}\begin{array}{rcl}& & 胃\end{array}\begin{array}{rcl}& & \sin \end{array}\begin{array}{rcl}& & 蠒\end{array}\begin{array}{rcl}& & \stackrel{鈭�}{胃}\end{array}\begin{array}{rcl}& & +\end{array}\begin{array}{rcl}& & \cos \end{array}\begin{array}{rcl}& & 蠒\end{array}\begin{array}{rcl}& & \stackrel{鈭�}{蠒}\end{array}$

.

As $x=\left(r\sin 胃\right)\cos 蠒$, $y=\left(r\sin 胃\right)\sin 蠒$, $z=r\cos 胃$, the position vector is

$\stackrel{}{r}=\left(r\sin 胃\right)\cos \stackrel{鈭�}{x}+\sin 胃\sin 蠒\stackrel{鈭�}{y}+\cos 胃\stackrel{鈭�}{z}$

Multiply above equation by $\cos 胃$on both sides,

$\cos 胃\stackrel{鈭�}{r}=\sin 胃\cos 胃\cos \stackrel{鈭�}{x}+\sin 胃\cos 胃\sin 蠒\stackrel{鈭�}{y}+{\cos }^2胃\stackrel{鈭�}{z}$ 鈥︹��. (6)

Now the theta vector is $\stackrel{鈭�}{胃}==\cos 胃\cos 蠒\stackrel{鈭�}{x}+\cos 胃\sin 蠒\stackrel{鈭�}{y}-\sin 胃鈥�\stackrel{鈭�}{z}$

Multiply above equation by $\sin 胃$on both sides,

$\sin 胃\stackrel{鈭�}{胃}=\sin 胃\cos 胃\cos 蠒\stackrel{鈭�}{x}+\sin 胃\cos 胃\sin 蠒\stackrel{鈭�}{y}-{\sin }^2胃\stackrel{鈭�}{z}$ 鈥︹��. (7)

Subtract equation (7) from equation (8) as,

$\begin{array}{rcl}\cos 胃\stackrel{鈭�}{r}-\sin 胃\stackrel{鈭�}{胃}& =& \left[\sin 胃\cos 胃\cos \stackrel{鈭�}{x}+\sin 胃\cos 胃\sin 蠒\stackrel{鈭�}{y}+{\cos }^2胃\stackrel{鈭�}{z}-\sin 胃\cos 胃\cos 蠒\stackrel{鈭�}{x}-\sin 胃\cos 胃\sin 蠒\stackrel{鈭�}{y}+{\sin }^2胃\stackrel{鈭�}{z}\right]\\ & =& {\cos }^2胃鈥�\stackrel{鈭�}{z}+{\sin }^2胃鈥�\stackrel{鈭�}{z}\\ & =& \stackrel{鈭�}{z}\\ & & \end{array}$

Thus, $\stackrel{鈭�}{z}=\cos 胃\stackrel{鈭�}{r}-\sin 胃\stackrel{鈭�}{胃}$