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l_notes12 - Lecture XII Terminology for PointSets in...

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Lecture XII Terminology for Point-Sets in Euclidean Spaces and Minimum-Maximum Theorems First let us take a short look at a problem that was on the exam. We are given a level curve (in E 2 ) or a surface(in E 3 ) and a point P on that curve or surface. How to find a vector normal to that curve or surface at that point P ? Let us consider the case of E 3 . The answer lies in the gradient of the function defining the surface. The graph of the surface is given by z = f ( x, y ), hence z f ( x, y ) = g ( x, y, z ) = 0. The gradient of g at P is a vector normal to the surface at P : ∂f ∂f P = P j + ˆ ˆ k. g | ∂x | P ˆ i ∂y | Now let us take a look at some basic notions of point-set topology of Eu- clidean spaces. Definition 1 Given a point P , we define a neighborhood of P in the following manner: (i) For the 1-dimensional space, a neighborhood is an interval of the form [ c r, c + r ] for some r > 0, where c is the point P .
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