MUMBAI, India -- Off the injured list and back on the world stage, U.S. Secretary of State Hillary Rodham Clinton on Saturday gave an impassioned defense of American demands that India and other countries do more to tackle terrorism and global warming.

Opening a three-day visit to India, Clinton sought to emphasize common interests, symbolized by the terrorist attacks in this seaside city last November that killed 166 people. "It must be stopped," she said, adding that the United States cannot do it alone.

Part of the backdrop to Clinton's visit is a sense of unease among Indians that the Obama administration is focusing more on its anti-terror campaigns in Afghanistan and Pakistan, at the expense of attention to the world's largest democracy.

Clinton is the highest-ranking administration official to stop in India, where she has been widely popular since visiting as first lady in the 1990s.

Showing no visible effects from elbow surgery in mid-June, Clinton met with business leaders, was serenaded by female entrepreneurs and participated in a televised discussion at a college on what's wrong with education in India and the U.S.

It was the first event of her day, however, that underscored most strongly the central message she carried from Washington.

"The bottom line for me is, our government is committed in the fight against terrorism," she told reporters at the Taj Mahal Palace and Tower after meeting privately with a group of survivors of last year's attacks on the Taj and an adjacent hotel.

"And we expect everyone" who shares the American desire to end violent extremism "to take strong action to prevent terrorism from taking root on their soil and making sure that terrorists are not trained and deployed — and we believe that around the world."

Her voice rising, she noted the lasting effects from the Sept. 11 attacks and said no one should doubt U.S. resolve.

"We are fighting wars to end the threat of terrorism against us, our friends and allies around the world," she said at a poolside patio at the Taj that had been strewn with bodies after last year's attack. She indicated that in New Delhi, the capital where her trip continues Sunday, she would speak with government officials about how India can play a stronger role against terrorism.

In an event closed to reporters, Clinton met at the Taj with workers who survived the attacks. One was the hotel's general manager, Karambir Kang, who lost his wife and two children during the three-day siege.

In a memorial book, she wrote: "Americans share a solidarity with this city and nation. Both our people have experienced the senseless and searing effects of violent extremism. And both can be grateful and proud of the heroism of brave men and women whose courage saved lives and prevented greater harm" last November and on Sept. 11, 2001. "Now it is up to all nations and people who seek peace and progress to work together. Let us rid the world of hatred and extremism that produces such nihilistic violence."

One of the most contentious issues between India and the U.S. is the administration's push for India to accept limits on carbon emissions as part of an international climate change agreement. To emphasize the importance of the matter, Clinton traveled with the special U.S. envoy for climate change, Todd Stern.

Clinton urged India to take a leading role and to avoid the missteps that occurred during U.S. industrialization.

"We acknowledge now with President Obama that we have made mistakes in the United States, and we along with other developed countries have contributed most significantly to the problem that we face with climate change," she said. "We are hoping a great country like India will not make the same mistakes."

Obama said in a statement in Italy this month that the U.S. had "sometimes fallen short" of its responsibilities in controlling its carbon emissions.

Clinton also met with 11 business leaders, including Mukesh Ambani, chairman of Reliance Industries, the largest privately held company in India. She told them she agreed that different countries ought to approach climate change in different ways. But she also said, "There does have to be a framework that India and China in particular sign on to that produces results."

Echoing remarks made by Ambani at the meeting, Clinton told reporters later that India should leapfrog the developed world to come up with its own innovative way to encourage environmentally friendly growth.

"We believe India is innovative and entrepreneurial enough to figure out how to deal with climate change while continuing to lift people out of poverty and develop at a rapid rate," she said.