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The Space R
The Differential and Derivatives
3 other sections not shown
1-manifold algebra bijection C1 function C1 map chain rule containing continuous maps Corollary critical point curve Define g defined by f Definition denoted derivatives upto order differentiable everywhere directional derivative equations f is continuous f is differentiable f is locally finite sum g o f given Hence f implicit function theorem inverse function theorem inverse map Jacobian matrix Let f lim f(x linear map locally invertible manifold map f map g maximum mean value theorem minimum neighbourhood nonzero Note obtain open ball open interval open set open subset partial derivatives upto polynomial positive real number Proof Proposition 2.2 Proposition Let Proposition Suppose R-+R rational function repeated partial derivatives required identity result follows Rn and f Rn-+R Rn+m RnxRm S-+Rm satisfies scalar multiplication set of Rn subset of Rn Suppose that f Taylor's expansion Taylor's theorem unit vector y e Rn zero map