cal15 - Chapter Fifiteen Surfaces Revisited 15.1 Vector...

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15.1 Chapter Fifiteen Surfaces Revisited 15.1 Vector Description of Surfaces We look now at the very special case of functions r D R 3 : , where D R 2 is a nice subset of the plane. We suppose r is a nice function. As the point ( , ) s t D moves around in D , if we place the tail of the vector r ( , ) s t at the origin, the nose of this vector will trace out a surface in three-space. Look, for example at the function r D R 3 : , where r i j k ( , ) ( ) s t s t s t = + + + 2 2 , and D R 2 = - {( , ) : , } s t s t 1 1 . It shouldn't be difficult to convince yourself that if the tail of r ( , ) s t is at the origin, then the nose will be on the paraboloid z x y = + 2 2 , and for all ( , ) s t D , we get the part of the paraboloid above the square - ≤ 1 1 x y , . It is sometimes helpful to think of the function r as providing a map from the region D to the surface. The vector function r is called a vector description of the surface. This is, of course, exactly the two dimensional analogue of the vector description of a curve.
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15.2 For a curve, r is a function from a nice piece of the real line into three space; and for a surface, r is a function from a nice piece of the plane into three space. Let's look at another example. Here, let r i j k ( , ) cos sin sin sin cos s t s t s t t = + + , for 0 t π and 0 2 s π . What have we here? First, notice that | ( , )| (cos sin ) (sin sin ) (cos ) sin (cos sin ) cos sin cos r s t s t s t t t s s t t t 2 2 2 2 2 2 2 2 2 2 1 = + + = + + = + = Thus the nose of r is always on the sphere of radius one and centered at the origin. Notice next, that the variable, or parameter, s is the longitude of r ( , ) s t ; and the variable t is the latitude of r ( , ) s t . (More precisely, t is co-latitude.) A moment's reflection on this will convince you that as r is a description of the entire sphere. We have a map of the sphere on the rectangle
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15.3 Observe that the entire lower edge of the rectangle (the line from ( , ) 0 0 to ( , ) 2 0 π ) is mapped by r onto the North Pole, while the upper edge is mapped onto the South Pole. Let r D ( , ), ( , ) s t s t be a vector description of a surface S , and let p r = ( , ) s t be a point on S . Now, c r ( ) ( , ) s s t = is a curve on the surface that passes through he point p . Thus the vector d ds s s t c r = ( , ) is tangent to this curve at the point p . We see in the same way that the vector r t s t ( , ) is tangent to the curve r ( , ) s t at p .
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15.4 At the point p r = ( , ) s t on the surface S , the vectors r s and r t are thus tangent to S .
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This note was uploaded on 01/29/2012 for the course ENG 2zz3 taught by Professor Proff during the Spring '11 term at McMaster University.

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cal15 - Chapter Fifiteen Surfaces Revisited 15.1 Vector...

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