微积分教学课件chapter12.ppt

In this chapter we introduce vectors and coordinate systems for three-dimensional space. This will be the setting for our study of the calculus of functions of two variables in Chapter 14 because the graph of such a function is a surface in space. In this chapter we will see that vectors provide particularly simple descriptions of lines and planes in space. Three-Dimensional Rectangular Coordinate Systems The Cartesian product is the set of all ordered triples of real numbers and is denoted by . We have given a one-to-one correspondence between points P in space and ordered triples (, b, c)in Distance Formula in Three Dimensions The distance between the points and  is Equation of a Sphere An equation of a sphere with center (h,k,l) and radius r is  .In particular, if the center is the origin O, then an equation of the sphere is In three-dimensional analytic geometry , an equation in x, y, and z represents a surface in . For example: 12.6 Cylinders and Quadric Surfaces A cylinder is a surface that consists of all lines that are parallel to a given line and pass through a given plane curve. Quadric Surfaces A quardic surface is the graph of a second- degree equation in three variables x, y, and z. The most general such equation is ilips?id]? 1.ellipsoid 2.Elliptic paraboloid iliptik] p?r?b?l?id]? 3.Hyperbolic paraboloid Theorem If is the angle between the vectors and ,then Properties of the cross product Proof From the definitions of the cross product and length of a vector,we have Corollary Two nonzero vectors and are parallel if and only if The length of the cross product is equal to the area of the parallelogram determined by and Example Find a vector perpendicular to the plane that passes through the points and P