Describe in words the surface whose equation is given.phi=pi/3

Solutions

VerifiedFrom the given $\phi$ we notice that $0\lt\phi\lt\frac{\pi}{2}$, therefore the given equation represents a half-cone.

The spherical coordinate points of $P$ are

$\left(\rho, \ \theta, \ \phi\right), \quad \rho>0, \quad 0<\phi<\pi$

where $\rho$ represents the distance from the origin to $P$, $\theta$ denotes the same angle as in cylindrical coordinates, and $\phi$ is the angle between the positive $z$-axis and the line segment $OP$.

To convert from spherical to rectangular coordinates we use the following equations:

$\begin{align*} x&=\rho \sin \phi \cos \theta, \\ y&=\rho \sin \phi \sin \theta, \\ z&=\rho \cos \phi. \end{align*}$

$\begin{align*} \phi&= \pi/3\\ \cos \phi&= \cos(\pi/3) = \dfrac{1}{2}\\ \cos^2 \phi&=\dfrac{1}{4}\\ \rho^2 \cos^2 \phi&=\dfrac{1}{4}\rho^2\\ z^2 &= \dfrac{1}{4}(x^2 + y^2 +z^2) \\ 4z^2 &= x^2 + y^2 +z^2\\ 3z^2 &= x^2 + y^2 \end{align*}$

$3z^2 = x^2 + y^2$ is a double cone, however, since the original equation $\phi= \pi/3 > 0$, it's the upper half of the double cone.

Spherical coordinates:

(1) $x = \rho \sin \phi \cos \theta$

(2) $y = \rho \sin \phi \sin \theta$

(3) $z = \rho \cos \phi$

(4) $x^2 + y^2 + z^2 = \rho^2$

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