Introduction to Calculus in Several Variables

1 2 Contents 0. One-variable calculus 1. The derivative 2. Inverse function and implicit function theorem 3. Fundamental local existence theorem for ODE 4. The Riemann integral in n variables 5. Integration on surfaces 6. Differential forms 7. Products and exterior derivatives of forms 8. The general Stokes formula 9. The classical Gauss, Green, and Stokes formulas 10. Holomorphic functions and harmonic functions 11. The Brouwer fixed-point theorem A. Metric spaces, convergence, and compactness B. Partitions of unity C. Differential forms and the change of variable formula D. Remarks on power series E. The Weierstrass theorem and the Stone-Weierstrass theorem 3