President Carter has registered dramatic gains in the wake of the last week's Democratic National Convention and has now drawn even with Ronald Reagan, according to the latest Gallup poll.

Carter also registered sharp gains in two other polls taken after the convention, but still trails Reagan in both.

The new Gallup figures show Reagan with 39 percent, Carter with 38 percent and independent candidate John B. Anderson with 14 percent. This, according to George Gallup, means that Regan and Carter "are now in a dead heat" -- a marked contrast to the 14-point lead that Reagan held over Carter in a Gallup survey a week before the Democratic convention.

The latest Gallup poll also demostrates convincingly that Anderson's independent candidacy is far more damaging to Carter than to Reagan.

When Anderson's name was omitted from the questions in this post-convention survey, Carter jumped into a six-percentage-point lead, with 46 points to Reagan's 40. Thus, the addition of Anderson's name in the three-way sampling caused Carter to lose eight points, while Reagan lost just one.

The latest Louis Harris survey shows Reagan with a six-point lead, as compared with a 20-point lead before the Democratic convention. Harris' figures showed Reagan with 42, Carter with 36 and Anderson with 17.

The Harris survey has consistently shown Reagan with larger leads over Carter than has the Gallup poll. The Harris survey taken just before the Democratic convention seemed to buttress the claims of the Kennedy forces that Carter's nomination would be disatrous for Democrats in the fall. In that Harris sampling of "likely voters," Democrats outnumbered Republicans by just 7 percentage points. This was a margin significantly smaller than that of Democrats to Republicans used by other polling organizations. But in the latest Harris survey, according to a Harris representative, the sampling of likely voters increased to 16 percentage points more Democrats than Republicans.

Another survey, conducted Friday and Saturday for the Associated Press and NBC showed Carter still trailing Reagan, but by only seven points, as opposed to 25 points in a sampling taken before the Democratic convention. In the latest AP-NBC poll, Reagan had 39 percent, Carter 32, and Anderson 13.

In the latest Gallup sampling, based on registered voters, Democrats outnumbered Republicans by 29 percentage points.

The Gallup poll, conducted from Friday through Sunday, showed a significant increase in the number of persons now saying they approve of the way Carter is handling his job as president. In the post-convention survey, 31 percent said they approved, as opposed to 21 percent in the preconvention poll. But 55 percent still said they disapproved of the way Carter is handling his job, which is majority of those questioned, but is significantly less than the 63 percent who said they disapproved in the questioning earlier this month.

The latest round of polling produced similar reactions from the Carter and Reagan camps.

"This is clearly the post-convention push that presidential nominees get," said Carter press secretary Jody Powell. "The Republicans got theirs and now we've gotten ours."

Meanwhile, Reagan, campaigning in Philadelphia, said: "It is natural that after a convention there is an upsurge. There was for me after the Republican convention. Now there is for him."

But the two camps parted company on the question of what the poll results would be the next time around.

"They are going to find out, just like we did in 1976, that this between conventions euphoria quickly disapears and is never seen again," said Powell, predicting a close contest from now on. "Dreams of carrying 45 states on Election Day quickly degenerate into a fight for the basic minimum number of electoral votes needed to win."

Carter pollster Patrick Cadell had said, during the Democratic convention, that he expected Carter to be 10 to 12 percent points behind Reagan by mid-September and then to overtake Reagan as the campaign progressed. He said there is a rhythm to presidential campaigns and -- acknowledging that it sounded ridiculus to be saying so at a time when the polls showed Carter still trailing badly -- Caddell said there was danger that Carter would come on too fast and "peak" too soon.

Meanwhile, Joe Holmes, Reagan's director of communications, said that he thinks Carter's gains will prove transitory. He said that the Reagan campaign pollig, conducted by Richard Wirthlin, showed that Carter had made gains at the Democratic convention but that Reagan still holds the lead.

"We think Mr. Carter may not continue in any more of a rise," Holmes said. When the next round of polls is made public, he added: "We think we'll look pretty good again."

John Anderson, meanwhile, said that he was not worried about his 14 percent showing in the latest Gallup poll -- a matter which could cause him concern, since the League of Women Voters has said that independent candidates will need to average 15 percentage points in the polls in order to qualify for its presidential debates.

"I'm not worried at all," Anderson said. "There's a new Harris poll which give me 17 percent."