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Unformatted text preview: Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a ∈ S where S looks smooth , i.e., without any fold or cusp or self-crossing , we can intuitively define the tangent plane to S at a as follows. Consider a plane Π which lies outside S and bring it closer and closer to S till it touches S near a at only one point, namely a , without crossing into S . This intuitive picture is even clearer in the case of finding the tangent line to a smooth curve in the plane at a point. A slightly different way is to start with a plane Π in R 3 which slices S in a small neighborhood U = B a ( r ) of a , and then consider the limit when Π ∩ U shrinks to the point a , and call the limiting plane the tangent plane at a . In the case of a plane curve arising as the graph of a nice function f , one knows that this process works and gives the slope of the tangent line to be f ( a ). In higher dimensions, if S lies in R n , one can define tangent vectors by considering smooth curves on S through the point a , and when S is a level set of the form f ( x 1 ,...,x n ) = c , one can use the gradient of f at a to define the tangent space. But this method fails when the gradient vanishes at a . Finally, it should be noted that one often defines the tangent space at a , when it makes sense, in such a way that it becomes a vector space. But if one looks at the case of plane curves, the usual tangent line is not a vector space as it may not contain the origin 0 (= (0 , 0)), unless of course if a = 0. It becomes a vector space if we parallel translate it to the origin by subtracting a . It is similar in higher dimensions. One calls a plane in 3-space an affine plane if it does not pass through the origin. 1 3.1 Tangents to parametrized curves By a parametrized curve , or simply a curve , in R n , we mean the image C of a continuous function α : [ r,s ] → R n , where [ r,s ] is a closed interval in R . C is called a plane curve , resp. a space curve , if n = 2, resp. n = 3. An example of the former, resp.the latter, is the cycloid , resp. the right circular helix , parametrized by α : [0 , 2 π ] → R 2 with α ( t ) = ( t- sin t, 1- cos t ), resp. β : [- 3 π, 6 π ] → R 3 with β ( t ) = (cos t, sin t,t ). Note that a parametrized curve C has an orientation , i.e., a direction ; it starts at P = α ( r ), moves along as t increases from r , and ends at Q = α ( s ). We call the direction from P to Q positive and the one from Q to P negative. It is customary to say that C is a closed curve if P = Q . We say that C is differentiable iff α is a differentiable function. Definition. Let C be a differentiable curve in R n parametrized by an α as above. Let a = α ( t ), with t ∈ ( r,s ). Then α ( t ) is called the tangent vector to C at a (in the positive direction)....
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This note was uploaded on 04/18/2008 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.
- Spring '08