Contributed by
Christopher Wolfe
One must wonder how aliens might communicate with humans when and if they arrive on Earth. In the 1951 film The Day the Earth Stood Still, the extraterrestrial Klaatu (Michael Rennie) introduces himself to the genius Professor Barnhardt using the universal language of math. After sneaking into the professor’s office and examining a chalkboard full of what appears to be very basic differentiation and integration, Klaatu pinpoints the professors flaw, corrects the equation, and waits for Barnhardt’s response. Following the awkward introductions, Klaatu reveals his expertise in the equation on the board, since this very problem in “celestial mechanics” allowed him to travel to Earth.
The math is later set aside to reveal the true purpose of the story. Klaatu’s mission is not to enlighten men with mathematics, but to warn them against nuclear programs. The primitive scientists have no need for such destructive weapons, so they must disarm in order to prevent the destruction of Earth.

There is really not much math in this movie, but the bit that is there is handled nicely. I like the reference to variation of parameters particularly:
(quoted from The Day the Earth Stood Still)
Copied from scifiscripts.com where the script seems to be available for free:
BARNHARDT
You wrote this?
KLAATU
(nodding easily)
It was a clumsy way to introduce
myself  but I understand you're a
difficult man to see.
(glancing at the
blackboard
reproachfully)
I thought you'd have the solution by
this time.
BARNHARDT
Not yet. That's why I wanted to see
you.
Klaatu glances at the work Barnhardt has been doing on the
board. Then he points to one of the expressions in an
equation.
KLAATU
All you have to do now is substitute
this expression
(pointing to a specific
place)
at this point.
Impressed and interested, Barnhardt tugs at his chin as he
studies and weighs the results.
BARNHARDT
(slowly, thoughtfully)
Yes  that will reproduce the first
order terms. But what about the effect
of the other terms?
KLAATU
Almost negligible... With variation
of parameters, this is the answer.
BARNHARDT
How can you be so sure? Have you
tested this theory?
KLAATU
(with a slight smile)
I find it works well enough to get
me from one planet to another.

Contributed by
Brett Wormley
I saw this movie as a young teenager, and thoroughly enjoyed it. I usually pay attention to details I don't understand, like new words or concepts. I mentally logged "variation of parameters" and wondered what it meant, but thought it was just fiction. Ten years later, in my ordinary differential equations class, I finally learned the technique, and told the professor I had been waiting for a decade to understand what Klaatu meant!

An anonymous contributor wrote to point out that Sam Jaffe, who plays the Einsteinlike scientist in the film, actually worked as a math teacher prior to becoming an actor and may have played a role in formulating the equations on Barnhardt's blackboard.
