AS A RESULT of Tuesday's vote, top offices in the District as well as in Prince George's, Montgomery and Arlington counties will be under new management. Some of the most important challenges confronting these leaders - public transportation, air and water problems, housing needs and so on - are not confined to narrow jurisdictions. So it will not be enough for them to give the usual lip-service to "regional cooperation." How efficiently and amiably the new leaders of the District and the surrounding counties work together on serious regional business will have a profound effect on the welfare of everybody who lives in the Greater Washington area.

To an extraordinary degree, the local voting turned not so much on ideology or party politics as on quality of management. Thus a heavily Democratic electorate in Prince George's could turn comfortably to a Republican to be county executive; in Montgomery, the switch could be from a Republican to a Democrat; in Arlington, the Democrats could split their ticket to give the county board a Republican majority. In the District, the race was decided in September's Democratic primary, and the Democrats clearly voted for better management when they abandoned Walter Washington, 2-to-1, in favor of a change. Indeed, management was the most serious issue raised by Republican Arthur Fletcher in his forlorn campaign against Marion Barry.

Obviously, the first things the area's new leaders need to work out are their own individual management problems. Marion Barry's first priority, in our view, is to find the best city administrator available, as he promised to do - an experienced technician. With a new team of department heads assembled, preliminary work can go forward on a legislative agenda for 1979, including such urgent regional concerns as the financing of Metro.

The matter of Metro also happens to be the principal regional concern in Arlington, where at least one of the three county board members has been an impassioned critic of the system's operations and an advocate of curtailing the rail system. The newest member, Stephen H. Detwiler, has promised to keep property taxes at the same dollar level next year as this year, which would require either a 5- to 10-percent cut in the budget to cover the costs of inflation or another source of revenue. How these concerns will be relected on the regional Metro board remains to be seen.

The challenges of making divided government work may be most acute in Prince George's, where the voters chose Republican Lawrence J. Hogan for county executive but returned a solidly Democratic council. Mr. Hogan's choices for key administrative jobs, and the council's treatment of his nominees, will be an early test of how well this relationship is going to work - and of how serious Mr. Hogan is about his commitment to bring top-quality new people, including more blacks and women, into county government. The new management is Upper Marlboro also faces a host of fiscal issues, including Metro financing, that are bound to be complicated by the freezing on property-tax revenues that county voters approved Tuesday.

The transition in Montgomery County should involve fewer shocks and strains. Since executive James P. Gleason is retiring, transition planning has been under way in Rockville for some time. Democrat Charles W. Gilchrist, who won the county executive race comfortably, will be working with a Democratic council - and will have more fiscal flexibility, since the voters wisely rejected the Montgomery version of TRIM. Still, Mr. Gleason has left Mr. Gilchrist a number of problems with large regional implications, starting with the matter of sewage treatment.

Most of the new figures in charge in the area will have plenty to do at the start in their own bailiwicks. But there is much that can only be accomplished regionally, in concert. The new managers in the Greater Washington area should make it a point to get to know each other - the sooner the better.