《高数双语》课件section 8.4.pptx

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1、Section 8.4,Surfaces and Space Curves,1,Julius Plcker,2,Equations for Surface,The curve can be thought as a trace of a variable point.,Similarly,a surface can also be regarded as the generating trace of a variable,point(or a movable curve)that moves according to some given law.,3,given conditions.,I

2、f P moves in S,from those conditions,we can obtain,and equation in the three variables x,y and z,It is clear that if the point P lies on S,then its coordinates,in the rectangular,Suppose that the coordinates of P are,must satisfy this equation;,conversely,if the coordinates x,y,and z,of the point P

3、satisfy this equation,then the point P must lie on S.,Thus,the equation(1)is called the equation of the surface S,and,(1),the surface S is called the Figure of the equation.,Equations for Surface,4,When graphing a cylinder or other surface by hand or analyzing one,generated by a computer,it helps to

4、 look at the curves formed by,intersecting the surface with planes parallel,to the coordinate planes.,These curves are,called cross sections or traces.,Equations for Surface,5,the distance from P to P0 is R.,Therefore the equation of the sphere,S in rectangular coordinate is,Equations for Surface,6,

5、Cylinders(柱面),7,The Parabolic Cylinder y=x2,Solution,Suppose that the point,lies on the parabola,in the xy-plane.,Then,for any value of z,will lie on the cylinder,the point,parallel to the z-axis.,because it lies on the line,through,Conversely,lies on the,any point,cylinder.,The equation of this cyl

6、inder is y=x2.,Finish.,8,Cylinders,It is easy to see that a cylinder is determined uniquely by its directrix,and a fixed line C,but the directrix of a cylinder is not unique.,Therefore,the equation of S is,9,Their figure is shown in the following.,Cylinders,10,Cylinders,11,Cones(锥面),12,generating li

7、ne OP,and OP crosses,It is not difficult to see that,the coordinates of points P and P0 satisfy,the following equations:,Cones(锥面),at,13,and so,This is the equation of the cone whose vertex is,the origin and whose directrix is the curve,in the plane,Cones(锥面),14,Elliptic Cone,For example,suppose S i

8、s a cone whose vertex is the origin,That is,The equation of the cone S is,This is called an elliptic cone.,15,Graphing Cones,The elliptic cone,is symmetric with respect to the three coordinate planes.,the point,the lines,the lines,Then the graphic of this elliptic cone can be shown in the following.

9、,16,Graphing Cones,17,Surfaces of Revolution（旋转曲面）,18,obtained from the point,by the revolution,so that,from M to z-axis,and hence,in S,is equal to the distance,Surfaces of Revolution（旋转曲面）,19,the last equation and this equation is the equation of the surface of revolution.,Surfaces of Revolution（旋转

10、曲面）,20,Surfaces of Revolution,The equations of the surfaces formed by the parabola,revolving around the z-axis is,This is called a paraboloid of revolution.,21,The equations of the surfaces formed by the hyperbola,revolving around the x-axis is,This is called a hyperboloid of,two sheets of revolutio

11、n.,Surfaces of Revolution,22,The equations of the surfaces formed by the hyperbola,revolving around the y-axis is,This is called a hyperboloid of,one sheets of revolution.,Surfaces of Revolution,23,Quadric Surfaces(二次曲面),A surface is called a quadric surface,if it can be represented by a,general for

12、m is,simplified by translation and rotation,as in the two-dimensional case.,The basic quadric surfaces are ellipsoids,paraboloids,elliptic cones,24,Quadric Surfaces（二次曲面）,The equation of an ellipsoid(椭球面),The equation of an elliptic paraboloid(椭圆抛物面),The equation of a hyperbolic paraboloid(双曲抛物面),Th

13、e equation of a hyperboloid of one sheet(单叶双曲面),The equation of a hyperboloid of two sheet(双叶双曲面),25,Graphing Ellipsoids,26,Graphing Paraboloids,27,Graphing a Saddle Surface(马鞍面),28,Graphing Hyperboloids,29,Graphing Hyperboloids,30,Space Curves,General form of equations of space curves,We know that

14、a straight line in space can be regarded as the line of intersection,regarded as the curve of intersection of two surfaces.,31,Parametric equations of space curves,A space curve can be represented by parametric equations in the,same way as a straight line in space.,Equations of Space Curves,32,Proje

15、ctions of Space Curves on Coordinate Planes,The cylinder whose directrix is and whose generating line is parallel,to the z axis is called a projecting cylinder of on the xOy plane.,The,curve of intersection of the projecting cylinder and the xOy plane is,called the projecting curve(or projection)of

16、on the xOy plane.,33,Let the general form equation of the space curve be,Projections of Space Curves on Coordinate Planes,34,Example Find the equations of the projecting curves of the curve,to the three coordinate planes,respectively.,Solution,C is a curve of intersection,of the upper hemisphere,var

17、iable z,the equation of projection to xOy,plane is just,and the circular cylinder,Projections of Space Curves on Coordinate Planes,35,of C on the xOy plane is,Projections of Space Curves on Coordinate Planes,36,Solution(continued),the xOy plane is,Similarly,eliminating the variable y from the equati

18、on C,we have,so that the equation of the projecting curve of C on,Finish.,Projections of Space Curves on Coordinate Planes,Cylindrical Coordinate System,37,Definition Cylindrical Coordinates,Cylindrical coordinates represent a point P,as the,in space by ordered triples,right figure.,Cylindrical Coor

19、dinate System,38,Example Find an equation for the saddle surface z=x2-y2 in cylindrical coordinates.,Solution,Finish.,Spherical Coordinate System,39,Definition Spherical Coordinates,Spherical coordinates represent a point P,as the,in space by ordered triples,right figure.,备注,40,课本上的球面坐标三个量的定义与课件上完全一致，不过写成分量形式时，一个点的表示是：也就是说把第二个量和第三个量换了位置。在做作业表示一个点时，请按照书上的顺序写。,Spherical Coordinate System,41,Example Find the equation in rectangular coordinates forthe surface.,Solution,Therefore,Finish.,