Judith Jackson, a senior at Duke University, has three goals she wants to accomplish this spring: passing final examinations, qualifying for entrance in a ''prestigious law school,'' and bestowing a legacy - an atmosphere of good race relations - on Duke, an institution that had no blacks 20 years ago.
She is one of two black student members - the other is Robert Harrington of Florence, S.C. - on the board of directors of a key project designed to improve race relations on campus, the new Mary Lou Williams Center for Black Culture.
The center, scheduled to open in September, is named after a noted jazz pianist, Mary Lou Williams, who also wrote sacred masses that have been performed in churches and in concert.
''We don't want the center to be a hangout just for black or minority students,'' Miss Jackson said. ''Our goal is to offer students, faculty, and community a place to study and understand the American black culture. We feel this will promote racial harmony in this Southern school.''
''We hope the Mary Lou Williams Center will have the most student traffic of any campus building,'' says Caroline Lattimore, vice-chancellor and assistant dean of minority affairs at Duke. ''It will be programmed with black cultural activities, from lectures to exhibits. If nonblacks don't come, the center would not achieve its goal of bringing all races together.''
In building the center, Duke hopes to forestall the kind of militant minority student discontent which erupts periodically on Northern campuses. At Harvard, for example, black law students are demanding more black and female faculty members; black students are not cooperating with the Harvard Foundation, set up to improve race relations after several years of protest by minority students and various coalitions; and black MBA students and alumni are seeking new means of retaining and enrolling minorities in the business school.
Duke, in an effort to retain and satisfy its black students, has set up various types of support programs - a preschool summer program; an Office of Minority Affairs; a special merit scholarship program, along with a general financial aid package; a Black Student Alliance; and a counseling program.
''We require all students to meet Duke entrance standards with no special breaks or status for minority students,'' Dr. Lattimore says. ''And we have set one basic goal for the new center: It will offer a cultural learning experience to all Duke students.''
William Griffith, vice-president of student affairs, adds, ''The black presence on our campus is accepted. It is not what we want it to be, but we are pledged to improve the situation.''
He lists five administration objectives:
* Increase black enrollment. ''We have no percentage or numerical goals.'' Currently 333 blacks are enrolled, nearly 6 percent of the total undergraduates.
* Increase the black faculty.
* Finance scholarship funds on a long-term basis. These include merit awards earmarked for black students. Endow them to increase in number and value.
* Provide a feasible financial aid package that fits the needs of black students, many of whom come from families with incomes of $12,000 or less, the annual cost of an education at Duke.
* Create a better atmosphere and climate in which black and white students live and learn together. The Mary Lou Williams Center is considered a major step in that direction.
''We all, black and white, work hard to recruit black students,'' Mr. Griffith said. ''Their biggest problems are financial and facing a predominantly white environment. It takes a unique young person to handle both issues.''
To North Carolinians, the administration stand is no surprise. Duke draws only 15 percent of its student body from the Tarheel State. President Terry Sanford, is a former governor. A liberal, he was the first governor ever to address the state's black Masonic Grand Lodge.
''He talks with you,'' Dr. Lattimore said of him. ''He works closely with us in the development of the center. He seeks improved race relations not only on campus, but throughout the state.''
President Sanford meets every three weeks with his own President's Council on Black Affairs, which includes key administrators in various departments. Dr. Lattimore's Office of Minority Affairs works closely with this group. Both Miss Jackson, a Bostonian who attended predominantly white suburban schools under a volunteer busing program, Metco (Metropolitan Council to Improve Education), and Mr. Harrington are among students working through Dr. Lattimore's office. Both are scholarship students - Harrington, an A.B. Duke scholar (the highest on campus), and Miss Jackson a Reginaldo Howard black scholar.
''I wish to be accepted as just another student at Duke, but not at the cost of giving up what is special about myself,'' Harrington said.
He says he believes the university sincerely wants more black students, but must provide recruiters more support, including endowment of the Reginaldo Howard Scholarship.
On campus, he says, blacks are split on what to do - concentrate on black activities, such as the Black Student Alliance, a black Greek letter society, and black activities in Durham, or go into the mainstream of campus activities. ''During my three years here we have constantly struggled to keep black activities going,'' Harrington said. ''At the same time we must deal with academic rigors and be part of the whole university family.''
''Coping at Duke goes beyond the traditional race relations between blacks and whites,'' Miss Jackson says. ''It includes relations among blacks themselves.
''Today, we have a new type of black enrolling at Duke, the black who is more affluent, who is not wholly dependent on student aid. And this new student does not always want to be identified as 'just another affirmative-action' black on campus.''