07Ma1cAnNotesChap 3

07Ma1cAnNotesChap 3 - Chapter 3 Tangent spaces normals and...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)....
View Full Document

{[ snackBarMessage ]}

Page1 / 12

07Ma1cAnNotesChap 3 - Chapter 3 Tangent spaces normals and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online