Pacific Ocean Asia) is in the Eastern Hemisphere, while the "East Pacific" (or "eastern Pacific", near the Americas) is in the Western Hemisphere. The Southern Pacific Ocean harbors the Southeast Indian Ridge crossing from south of Australia turning into the Pacific-Antarctic Ridge (north of the South Pole) and merges with another ridge (south of South America) to form the East Pacific Rise which also connects with another ridge (south of North America) which overlooks the Juan de Fuca Ridge. For most of Magellan's voyage from the Strait of Magellan to the Philippines, the explorer indeed found the ocean peaceful. However, the Pacific
Lie group Lie groups are called "isomorphic" if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras. The first result in this direction is Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says