Six months after mobs stoned the U.S. Embassy here in a rage over Washington's plan to recognize Peking, the new so-called unofficial U.S. relationship with Taiwan seems active and economically healthy, with few traces of bitterness toward Americans.

The new unofficial U.S. embassy -- called the American Institute in Taiwan -- has received a 30 percent jump in applications from Taiwanese seeking to visit the United States. Total Taiwan trade has increased 37 percent so far this year, and its trade with the United States remains far above the still meager commerce between China and the United States.

According to businessmen here, the new U.S. congressional legislation covering relations with Taiwan in some ways offers stronger guarantees to this island's future security than the mutual defense treaty that is scheduled to lapse at the end of this year.

"Congress was responsive to every major point we made," said Robert Parker, president of the American Chamber of Commerce in Taipei. As the cheif representative of U.S. business interests here, Parker may now be the most important American on the island.

While Taiwan government officials still say they are "not satisfied" with the new relationship, dissatisfaction does not appear to run deep. In recent weeks Taiwan officials have even been trying to persuade Japan and several European countries that do not maintain diplomatic links with Tapei to adopt the American formula.

Although the American Institute is allegedly a private organization, it uses the offices of the old U.S. military command and is staffed by U.S. Foreign Service officers on "temporary leave." The State Department employes have been assured that their salaries, benefits and seniority rights will continue and that they may immediately reenter official U.S. government service once they leave Taiwan.

The Institute and the corresponding Taiwan organization, the Coordinating Council for North American Affairs, are headed by experienced senior deplomats. The new American Institute director, and thus unofficial U.S. ambassador, is Charles Cross, who has taken what appears to be a genuine retirement from the State Department. Cross is fluent in Chinese and was formerly ambassador to Singapore and consul general in Hong Kong.

In the diplomatic tiptoe that has characterized the new triangular relations between Washington and Taipei and Washington and Peking, the American say relations with Taiwan are strictly unofficial while the Taiwanese say "qualities of officiality" remain. Neither side directly challenges the other's position. Both sides recognized the importance of the estimated $500 million American investments here and the fact that the American Institute, with a staff of about 50, is still the largest foreign representation in Taipei, official or otherwise.

The diplomats at the Institute do nearly all the same jobs they used to do at the embassy, including political and economic reporting back to Washington. Although the last U.S. military personel were officially withdrawn early this month, the Institute has a staff capable of handling Taiwan's purchases of U.S. military equipment.

A special U.S. facility set up in the hills north of Tapei to monitor military communications on the Chinese mainland is still operating. It is understood that Taiwanese now run the station, although there is some evidence that American civilians still; visit, at least. Some of the U.S. civilians remaining here reportedly handle intelligence operations.

Taiwan and American officials take heart from the language of the Taiwan Relations Act. It provides that any effort to determine the future of Taiwan by nonpeaceful means, including boycotts, embargoes or direct attack, would be of "grave concern to the United States" and require "appropriate action."

Yet it is the growing trade relationship that provides the most concrete guarantee of continued U.S. protection of this island of 17 million people. Following normalization, the U.S. Export-Import bank agreed to extend a $100 million credit to the Taiwan power company to finance the state-run utility development program. The Bank of America led a consortium of 19 banks in arranging a $99.7 million load package for the power company.

Americans living here now wrestle with the difficulties of maintaining a certain quality of life since the departure of many military families meant the loss of many amenities. The American School has lost a third of its pupils, and faculty cuts are under way.

Even more disastrous to some, the duty-free shopping privileges at the post exchange are no more. The American Institute is trying to fill the gap by opening a new liquor store. "Right now there are a lot of thirsty people here," one official said.