An obvious sense of growing frustration has begun to envelop President Carter's reelection campaign, which has stalled and so far failed to hit on a theme or tone that would throw Republican nominee Ronald Reagan on the defensive.

While Reagan has entered the final month of the presidential race in an apparently relaxed manner, the president's attacks on his opponent have grown sharper and his rhetoric more extravagant. Yet Reagan still clings to his steady lead and sloughs off the Carter attacks as the work of a desperate politician.

Carter aides insist that they are in no sense desperate and that Reagan remains a vulnerable candidate. "We still have enough to overtake him in the last week in October," a White House official said.

But they do not deny their frustration, which is beginning to take its toll, even on the president.

After a month of campaigning in which he sometimes seemed only to be going through the motions, Carter showed real emotion Monday night in Chicago while addressing a crowd of Democrats, some of whom were not bothering to listen.

Amid a stinging assault on Reagan on everything from arms control to unemployment compensation, the president let his frustration show as he said.

"It's difficult to get the message across to the American people in the hurly-burly of a campaign . . . . This is my last campaign, the last political race that I will ever run. I do not intend for it to end by turning the government of the United States over to people whose political philosophies and views about this country are directly contrary to everything in which I believe with all my heart and soul."

Carter's emotional appeal to the Chicago Democrats, who responded with lackluster applause and cheers, was the most striking example yet of the president momentarily dropping his public demeanor of calm and confidence. The first days of the campaign went well for him, but more recently the campaign has become stagnant while he remains behind. And issues and tactics that Carter strategists were confident would put Reagan on the defensive and focus the race on his qualifications for the presidency have so far not penetrated Reagan's defenses.

One of the president's problems is that he has enthusiastically adopted a campaign strategy that calls for him to portray Reagan as almost a political monster. The intent is to exploit public doubts about Reagan's qualifications for the job by suggesting, directily or otherwise, that he is a dangerous extremist who would turn back decades of social and economic progress and possibly plunge the country into war.

But often when Carter has tried to do this, attention has focused not on Reagan but on his own rhetoric. What attracted the most attention from reporters in Chicago Monday night, for example, was the president's suggestion that a Reagan victory could mean the separation of "black from white , Jew from Christian, North from South, rural from urban."

White House press secretary Jody Powell tried to defend this assertion by saying that Carter was deeply offended by Reagan's failure to condemn the Rev. Jerry Falwell, founder of the fundamentalist Moral Majority, for preasching that God hears the prayers of only "redeemed" Christians. In a visit to Falwell's headquarters in Lynchburg, Va., Oct. 3, Reagan said he disagreed with that philosophy, but he did not denounce it.

But Powell's explanation fell flat on its face because the president made exactly the same reference to the alienation of "Christian from Jew" in a speech to a Democratic fund-raising dinner in Washington Sept. 30, three days before Reagan's visit to Lynchburg. Powell did not know this because even he cannot keep up with all of Carter's attacks on his opponent.

Simialar efforts to find an issue to breathe life into the lagging campaign have also fizzled. After weeks of defensiveness on the question of the presidential debates, the Carter strategists thought they would get a boost from Reagan's refusal to engage in a two-man debate with the president. Carter mentions this often now, but he and his aides are extremely frustrated that it does not appear to be hurting the GOP nominee.

When the White House produced a list of past Reagan statements suggesting U.S. military involvement abroad, aides thought it would spark a round of sharp questioning of Reagan on foreign policy. But Reagan has ignored Carter's demand that he explain those statements, and once again the White House is frustrated.

"He is getting away with it," an official said. "They have clearly made a decision to moderate all of his positions. Reagan is now the great moderator instead of the great extremist of a few weeks ago. The press has not laid a glove on him . . .

"There are clearcut differnces, more so than in 1976, but we can't get the campaign focused on it. It is all politics and no substance. We are totally frustrated at our inability to get a debate going, at least at a distance, on some of these issues."