Math / Geometry

QuestionFind a parametrization of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ .

Studdy Solution

STEP 1

1. The sphere is centered at the origin with radius $a$ .
2. A sphere in 3D can be parametrized using spherical coordinates.
3. The parametrization should cover the entire surface of the sphere.

STEP 2

1. Introduce spherical coordinates for the parametrization.
2. Express the Cartesian coordinates $x$ , $y$ , and $z$ in terms of spherical coordinates.
3. Ensure the parametrization covers the entire sphere.

STEP 3

Introduce spherical coordinates $(\theta, \phi)$ where: - $\theta$ is the azimuthal angle measured from the positive $z$ -axis. - $\phi$ is the polar angle measured from the positive $x$ -axis in the $xy$ -plane.

STEP 4

Define the ranges for the spherical coordinates: - $\theta \in [0, \pi]$ - $\phi \in [0, 2\pi]$

STEP 5

Express the $x$ -coordinate using spherical coordinates: $x = a \sin(\theta) \cos(\phi)$

STEP 6

Express the $y$ -coordinate using spherical coordinates: $y = a \sin(\theta) \sin(\phi)$

STEP 7

Express the $z$ -coordinate using spherical coordinates: $z = a \cos(\theta)$

STEP 8

Combine the expressions for $x$ , $y$ , and $z$ to form the parametrization of the sphere: $\mathbf{r}(\theta, \phi) = (a \sin(\theta) \cos(\phi), a \sin(\theta) \sin(\phi), a \cos(\theta))$

STEP 9

Verify that the parametrization covers the entire sphere by varying $\theta$ from $0$ to $\pi$ and $\phi$ from $0$ to $2\pi$ .
The parametrization of the sphere $x^2 + y^2 + z^2 = a^2$ is: $\mathbf{r}(\theta, \phi) = (a \sin(\theta) \cos(\phi), a \sin(\theta) \sin(\phi), a \cos(\theta)), \quad \theta \in [0, \pi], \ \phi \in [0, 2\pi]$

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