## Calculus of Several Variables and Differentiable Manifolds |

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abstract algebraic arbitrary assume Banach space basis called Chapter choose closed components connected consider continuous convergent coordinates corresponding covariant cover critical point cube curvature curve defined definition depends derivative differential equation dimension directional element equivalent example exercise exists expression exterior derivative follows formula Frobenius function given Hence hypersurface independent initial values integrability conditions inverse Jacobian joining Lemma linear Lipschitz manifold matrix maximum Mean Value Theorem means metric MICHIGAN namely normal obtain one-to-one open set orientation orthogonal p-forms parameter partial plane positive definite problem Proof properties Prove rank relative Remark restricted result satisfied simply solution solve submanifold Suppose tangent vector tensor of type Theorem transformation unique unit UNIVERSITY OF MICHIGAN variables vector field vector space write written zero