Taylor's Theorem and proof
Taylor's Theorem and proof: Theorem : If f(x,y) is a function which possesses continuous partial derivatives of order n in any domain of a point (a,b) , and the domain is large enough to contain a point (a+h,b+k) with it, then there exists a positive number 0<θ<1, such that f(a+h,b+k) = f(a,b) + (h∂/∂x + k ∂/∂y)f(a,b) + (1/2!)(h∂/∂x + k∂/∂y)² f(a,b) + ....+{1/(n-1)!}(h∂/∂x + k∂/∂y)ⁿ⁻¹f(a,b)+Rₙ, where Rₙ = (1/n!)(h∂/∂x + k∂/∂y)ⁿf(a+θh,b+θk) 0<θ<1. proof: Let x = a+th , y = b+tk , where 0≤t≤1 is a parameter, and f(x,y) = f(a+th,b+tk) = φ(t) Since the partial derivatives of f(x,y) of order n are continuous in the domain under consideration , φⁿ(t) is continuous in [0,1], and also φ'(t) = df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) = h∂f/∂x + k∂f/∂y = (h∂/∂x+ k∂/∂y)f φ"(t) = (h∂/∂x + k∂/∂y)² f . . φⁿ(t) = (h