From “Ben Hur” to “Lord of the Rings,” Hollywood producers have long drawn upon – and stolen from – literary sources. In recent years, however, producers desperate to re-market known entities to shrinking, TiVo-enabled audiences have turned to cannibalizing kitschy television shows, video games, theme-park rides, and even – lowest of the low! – Broadway plays and musicals.

Before the year is out, theatergoers will have numerous opportunities to compare live productions to their celluloid offspring. Prompted by the runaway success of his Broadway debut, Mel Brooks has refilmed his 1968 comedy “The Producers” – this time with the key addendum “The Movie Musical.” The rock musical “Rent” will be released around Thanksgiving with most of its original cast intact, no doubt hoping to capitalize on the market for melismatic caterwauling that “American Idol” exploits. As for Off Broadway, there’s the movie version of David Mamet’s urban allegory “Edmond,” and Craig Lucas has adapted and directed his caustic look at Hollywood power games, “The Dying Gaul.” The first to the screens, however, is Miramax’s “Proof,” the mathcentric mystery-cum-romance that started as a Tony-winning Broadway play.

Conventional wisdom holds that a movie adaptation always will be inferior, the degraded byproduct of a mechanical process. If you’re a diehard fan of Patricia Highsmith, then Anthony Minghella’s picturesque but tame rendering of “The Talented Mr. Ripley” sent you into homicidal fits. Roland Joffe’s film version of “The Scarlet Letter” courted ridicule by inserting steamy sex scenes and Native American raids into the skeletonized plot of Nathaniel Hawthorne’s classic.

“Proof” the movie, though, is better than “Proof” the play. (A moment, please, as theater purists snort derisively.) “Proof” is a different story. Although I was familiar with and fond of the play when it premiered at the Walter Kerr Theatre five years ago, I found myself leaving the screening of the movie even more impressed than when I saw it on Broadway.

In both, the story takes place in Chicago over the course of a week. Robert (Anthony Hopkins) is a renowned mathematics professor at the University of Chicago who revolutionized the field in his 20s, but later in life suffered from mental instability (“Beautiful Mind”-style). Robert’s daughter, Catherine (Gwyneth Paltrow) is celebrating her 27th birthday as the story opens – although “celebrating” isn’t the right word. Depressed and bitter, Catherine has quit school to care for her deteriorating dad and is living in semi-squalor in her father’s house.

As the story unfolds, we learn that Robert recently died, although he still appears to Catherine with words of advice. Claire (Hope Davis), fearing that her sister might have inherited their father’s mental fragility, tries to convince Catherine to move to New York where she can be supervised. When Robert’s former student, Hal (Jake Gyllenhaal), discovers a groundbreaking proof in his late teacher’s voluminous notebooks, Catherine shocks everyone by claiming authorship of it.

On stage all the action took place on a back porch. It involved one setting and four characters, a lean and engrossing plot with witty dialogue and enough science to titillate without boring. To make a film out of it, playwright David Auburn and his co-screenwriter, Rebecca Miller, made some obvious interpolations: Scenes such as Robert’s funeral and the reception party afterward, for instance, were only alluded to onstage.

Flashbacks of Robert and Catherine’s troubled coexistence during his final years slide in and out of the main storyline through simple montage. The University of Chicago’s ivy-covered campus provides collegial exteriors, while studio interiors provide a wealth of detail to flesh out the house in which Robert and Catherine lived. This adds texture to the story and greater urgency to the juicy dualities – genius and madness, evidence and instinct, children and parents – central to the plot.

The burning question – did Catherine actually write that proof? – generates more suspense than it did on stage. Mr. Auburn and Ms. Miller concentrate on a detail that previously seemed minor: the fact that the notebook with the proof was locked in a drawer in Robert’s study. They use this central technique, crosscutting between scenes, to create a neat parallel between the father’s madness and the daughter’s creative breakthrough. The climax of the movie, which resolves these issues, thus has satisfying structural as well as emotional punch.

In terms of casting, there are bound to be theater- goers who saw Mary-Louise Parker in the role of Catherine and can’t imagine anyone else. But Ms. Paltrow does a credible job as the tortured young lady, bringing a softness and vulnerability somewhat lacking in Ms. Parker’s flinty, edgier rendition. There’s another factor, which can’t be avoided: age. Parker is a magnificent actress, but she’s 41, and the character’s age is crucially important. Catherine was 25 in the play, she was bumped up to 27 for the movie, and Ms. Paltrow was in her early 30s when filming started in 2003. Auburn emphasizes the notion that great mathematicians do their best work in their 20s. (Anyway, don’t shed tears for Ms. Parker – she’s doing quite well for herself in the Showtime series “Weeds.”)

Jake Gyllenhaal is less apt as Catherine’s love interest, Hal. Improbably hunky and pumped-up (was he simultaneously shooting “Jarhead”?), Mr. Gyllenhaal is simply not nerdy enough for the part. Hal is supposed to be a 21stcentury nerd, a symbol of “geek chic,” but he still must convey vestigial shyness and awkwardness. Hope Davis, on the other hand, is a perfect fit for the meddling, superficial Claire and adds sympathetic layers to the nearly thankless role. Anthony Hopkins rolls out his patented thoughtful gravitas to Robert. Even without an ideal cast (mine would be Maggie Gyllenhaal, Tobey Maguire, a dyed Hope Davis, and the original Robert, Larry Bryggman), “Proof” succeeds as a brainy, witty, and touching improvement on the play.

With so many plays on their way to the big screen, perhaps theater producers will worry that the filmed versions will upstage their originals. After all, if you’re more familiar with James Foley’s 1992 movie of “Glengarry Glen Ross” than the play itself, the jagged and terse original (recently revived on Broadway) might seem skimpier. The stage script also lacks the splendidly vicious monologue Mamet wrote for Alec Baldwin early in the film. And though Mike Nichols recently transferred Patrick Marber’s brittle sexual-infidelity drama “Closer” to film without adding much except London exteriors, he made his film debut with the abridged but much more potent film of “Who’s Afraid of Virginia Woolf?”

I prefer to look at these film adaptations another way: From one story sprang two outstanding works of art.