Finally, yesterday, the last preview party was over, the last special tour for notables, the last tune by the Marine Band, the last speech - and the East Building of the National Gallery of Art opened to the American public for which it was built.

The people had been waiting, at least 300 of them, in the roaring sun since 1, fanning themselves but cheerful, and when President Carter appeared they clapped and pressed forward and he shook their hands. By then the crowd had grwon to 2,000.

Paul Mellon, whose family money built the $94.4-million structure, introduced the president.

"We have no ministry of culture in this country," Carter said, "and I hope we never will. We have no official art, and I pray that we never will."

Standing squarely in front of what he called I.M. Pei's "architectural masterpiece . . . dignified and daring . . . monumental yet without pomposity," he saw the East Building as "a museum and as a center for art and scholarship" and "a metaphor for what, at its best, the relationship between art and government can be."

Then he toured the museum with gallery director J. Carter Brown, architect Pei, the Mellons and Joan Mondale.

He loved the David Smith room, a miniature amphitheater where he climbed among the huge iron statues and even leaned on one. He stopped before the immense glowering black painting by Robert Motherwell. But the best part was when he came to the mezzanine and found 100 staff members, caterers, carpentars, gardeners, and painters - marooned there by the Secret Service.

They whooped and cheered, and he shook some more hands and then went into the Picasso exhibit by himself. He spent about 20 minutes in the museum. Then the people got in.

During the previews, the visitors who roamed the six different levels looked like the stick figures in architectal drawings. But now, as the public poured in, fanned out and flowed up the staids and escalators, across the bridge and balconies to the very fartheast tower chamber, the scale of people to building seemed exactly right.

Even the three-story-tall Calder mobile, the artist's last completed work, was definitely turning faster. And the first fingermarks had appeared on the lowestvane, which makes a brief pass nine feet above the mezzanine.

From the bridge, 6-year-old Heather Lustig gazed at the jousting knights which celebrate the Dresden show and tried to be impressed. She was the first person into the building after the special people were admitted. Before that, President Carter had spotted her in the line and had tickled her and said, "Hi, Beautiful."

Nok a TV lady was asking for her impressions of the building and her mother, Diane, an Army wife here, encouraged her, too. "No," said Heather.

John Norris, a guard standing by a Brancusi sculpture, said he is moved to a different room every day, so that in about two weeks he can cover the whole building with its six major opening exhibits. At 44, a former careersoldier, he has been a guard three years.

There's always something new," he said. "I've learned a lot about art here met people from all over the world. Sometimes you stand in a room all day looking at some piece of art that doesn't mean a thing to you, and all of sudden you see what the guy was trying to say, where he was coming from. You get inside someone else's head."

By 4 p.m., nearly 3,000 people had come into the museum, which can handle 4,000 with ease. They took over. They moved slowly, mesmerized by the drama of the space, the scale. They snapped pictures of each other. They lined up at a counter to buy booklets. In the first hectic moments, many simply removed the booklets without paying, but soon the cash register was working just fine.

Already the system of passes was being used for the great Dresden exhibit: One picked up a free pass at a kiosk outside the old building to reserve space for oneself in one of the subsequent groups of people which pulse through the show at half-hour intervals.

"There's a lot of confusion today," said S.J. Vish, a lieutenant of guards. "This ticket thing is driving them all crazy. But we'll get it ironed out."

He was exhausted, he added. He had served until 1 a.m. at the final dinner party . . . had been up early planning tactics . . . wanted more guards for the staff.

Katherine Warwick, assistant to the director, was exhausted, too. "We all are," she said, "But its functioning! Its going! Its finally operating after eight years!"

In the superbly designed Dresden exhibit, people bunched up in the first of many chambers while they read their brochures.

"No, that's the traveling chest, and this is the desk over here," a woman told her companion.

They touched everything. Nose marks smudged the glass cases. An 18-century walnut table covered with chased and gilt silver had been protected by a thick glass cover. But the silver edge protruded an inch, and they touched that.

"I get so nervous they'll set off the alarm," remarked Curtiss Fennell, a guard in the gem room."They touch that glass hard enough, the alarm will go off."

He declined even to estimate how many times he had been asked if the jewels were real. They are so big you can't believe it, but they are.