Fundamental Ideas of AnalysisThe ideas and methods of mathematics, long central to the physical sciences, now play an increasingly important role in a wide variety of disciplines. Analysis provides theorems that prove that results are true and provides techniques to estimate the errors in approximate calculations. The ideas and methods of analysis play a fundamental role in ordinary differential equations, probability theory, differential geometry, numerical analysis, complex analysis, partial differential equations, as well as in most areas of applied mathematics. |
Common terms and phrases
approximate Bolzano-Weierstrass theorem C₁ Cauchy sequence choose coefficients complete complex numbers compute continuous function continuously differentiable continuously differentiable function converges uniformly countable define definition denote derivative difference quotient differential equation diverges Dom(f Euler's method Example exists Figure finite interval fn(x formula Fourier series func function ƒ Fundamental Theorem ƒ and g ƒ is continuous graph Hint implies iterates let ɛ Let f Let ƒ lim sup limit point linear Mean Value Theorem metric space N₁ Newton's method one-to-one partial sums pointwise Poisson random variable polynomials power series probability proof of Theorem properties Prove that ƒ radius of convergence real numbers Riemann integral Riemann sum satisfies Section sequence of functions series converges solution subinterval subset sup norm Suppose that ƒ t₁ tion triangle inequality upper bound