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1、Section 10.3,Application of Differential Calculus of Multivariable Function in Geometry,1,2,Overview,CURVE,SURFACE,1) Tangent line and normal plane,2) Tangent planes and normal lines,3,The Parametric Equations of a Space Curve,We already know that a plane curve can be represented by a parametric,by
2、a parametric equations, a line in space can be expressed,equations,or,of the variable point P(x,y,z).,4,The Parametric Equations of a Space Curve,Similarly, a space curve may also be represented by parametric,equations,or vector form,then is said to be a,on the interval,continuous curve;,If is a con
3、tinuous curve and,holds for any,and, then is said to be a simple curve.,5,The tangent line to ,The geometric meaning of the derivative of the direction vector r(t) at t0 is that r (t0 ) is the direction vector of the tangent to the curve at the corresponding point P0 .,r (t0 ) is called the tangent
4、vector to the curve at P0 .,O,x,y,z,T,The Vector equation of the tangent to the curve at P0 is,6,The equation of the tangent line to curve ,The Vector equation:,The Parametric equation:,The Symmetric equation:,7,The tangent line to ,A curve for which the direction of the tangent varies continuously
5、is called a smooth curve.,Example,piecewise smooth curve,8,The normal plane to ,We have seen that for a given space curve if r(t) is derivable at t0 and r(t0) 0, then the tangent to at P0 exists and is unique.,There is an infinite number of straight lines through the point P0 , which are perpendicul
6、ar to the tangent and lie in the same plane.,The plane is called the normal plane to the curve at P0.,through the point P0,perpendicular to the tangent,the equation of the normal plane,9,The normal plane to ,The equation of the normal plane to the curve at P0 is,Example Find the equations of the tan
7、gent line and the normal plane to the following curve at point t=1.,10,Tangent line and normal plane to a space curve,If the equations of the curve is given in the general form,and the above equations of the curve determine two implicit functions of one variable x, y=y(x) and z=z(x) in the neighbour
8、hood U(P0) and both y(x) and z(x) have continuous derivative. Then,the symmetric equation of the tangent at P0(x0,y0,z0) is :,11,Tangent line and normal plane to a space curve,and the equation of the normal plane at P0(x0,y0,z0) is :,12,2. Tangent planes and normal lines of surfaces,Normal line,Tang
9、ent plane,13,Parametrizing,Any space point can be imagined that,it lies on a sphere which is centered at,the origin and the radius is,If the angle between the projection vector,on the xOy plane and the positive,of,direction of x-axis is denoted by , and,and the positive direction of z-axis,the angle
10、 between the vector,is denoted by,then the two coordinate system are related by,14,Parametrizing,If we denote,the surface of,the angle between the projection vector,direction of x-axis is denoted by , and,Then the coordinate can,be expressed by,we can parametrize the equation of the curve or surface
11、.,15,Tangent Planes and Normal Lines to a Surface,Suppose that the parametric equation of a surface S is,tangent plane of any smooth curve on the surface through the point r0,16,Tangent Planes and Normal Lines to a Surface,Therefore, the normal vector is,Thus the tangent plane is,The normal line is,
12、17,Tangent Planes and Normal Lines to a Surface,Example Find the tangent plane and normal line to the right helicoid,at the point,18,Tangent Planes and Normal Lines to a Surface,Thus, the surface,and has continuous partial derivative.,S can be repressed by,It is easy to see that,then we have,or,19,T
13、angent Planes and Normal Lines to a Surface,The normal line is,20,Tangent Planes and Normal Lines to a Surface,Example Given an ellipsoid,and a plane,1) Find the tangent plane to the ellipsoid at the point P(x0,y0,z0) parallel to the plane .,2) Find the points on the ellipsoid with minimum and maximum distance to the plane .,21,Review,Tangent line and normal plane to a space curve Parametrizing The tangent plane and the norm line of a surface,
