Space Cases is about five space cadets, their two teachers and a malfunctioning android, Thelma. After the children see an alien spaceship, the Christa, outside their Space Academy and board it they become lost in space, with their assistant principal and teacher. Type: Scripted

Languages: English

Status: Ended

Runtime: 30 minutes

Premier: 1996-03-02

## Space Cases - Sobolev space - Netflix

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces, rather than in spaces of continuous functions and with the derivatives understood in the classical sense.

## Space Cases - Motivation - Netflix

The left-hand side of this equation still makes sense if we only assume u to be locally integrable. If there exists a locally integrable function v, such that

D                      α                          f        =                                                            ∂                                                      |                                    α                                      |                                                              f                                      ∂                              x                                  1                                                                      α                                          1                                                                                  …              ∂                              x                                  n                                                                      α                                          n                                                                                                          .              {\displaystyle D^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial x_{1}^{\alpha {1}}\dots \partial x^{\alpha _{n}}}}.}

∫                      Ω                          u                  D                      α                          φ                d        x        =        (        −        1                  )                                    |                        α                          |                                                ∫                      Ω                          φ        v                d        x        ,                φ        ∈                  C                      c                                ∞                          (        Ω        )        ,              {\displaystyle \int {\Omega }uD^{\alpha }\varphi \;dx=(-1)^{|\alpha |}\int \varphi v\;dx,\qquad \varphi \in C_{c}^{\infty }(\Omega ),}

where α a multi-index of order |α| = k and we are using the notation:

Ω              {\displaystyle \Omega }   is an open subset of                                           R                                n                          .              {\displaystyle \mathbb {R} ^{n}.}