北邮高等数学英文版课件Lecture 10-3

Section 10.3
Application of Differential Calculus of Multivariable Function in Geometry
1
Overview
CURVE
r( t ) x ( t )i y( t )j z ( t )k
z
SURFACE
F ( x, y, z ) 0
and the equation of the normal plane at P0(x0,y0,z0) is :
dy ( x x0 ) dx
dz ( y y0 ) dx x0
( z z0 ) 0
x0
Example Find the equations of the tangent line and the 2 x 2 y 2 z 2 45, normal plane to the curve at point P0(-2,1,6). 2 2 x 2y z
the symmetric equation of the tangent at P0(x0,y0,z0) is :
x x 0 y y0 z z 0 dy dz 1 dx x0 dx x0
10
Tangent line and normal plane to a space curve
14
Tangent Planes and Normal Lines to a Surface
Suppose that the parametric equation of a surface S is
r r( u, v ) ( x( u, v ), y( u, v ), z ( u, v )), ( u, v ) D R 2
x ( t ) x0 tx( t 0 ), y( t ) y0 ty( t 0 ), z ( t ) z tz( t ). 0 0
x x 0 y y0 z z 0 x ( t 0 ) y ( t 0 ) z ( t 0 )
The equation of the normal plane to the curve Γ at P0 is
x( t0 )( x x ( t 0 )) y( t 0 )( y y( t 0 )) z ( t 0 )( z z ( t 0 )) 0
Example Find the equations of the tangent line and the normal plane to the following curve Γ at point t=1.
where r is continuous in D, the point ( u0 , v0 ) D and the partial derivatives of r at the point ( u0 , v0 ) exist, that is,
z
r
P ( x( t ), y( t ), z( t ))
O
x
y
x
y
1) Tangent line and normal plane
2) Tangent planes and normal lines
2
The Parametric Equations of a Space Curve
We already know that a plane curve can be represented by a parametric
z
If the vector valued function r( t ) is continuous
on the interval [ , ], then Γ is said to be a continuous curve; If Γ is a continuous curve and
and r( t1 ) r( t 2 ) holds for any t1 , t 2 ( , )
y r sin sin , z r cos , 0 2 ,0 .
13
Parametrizing
Another way to parametrize is imagine that any point P ( x , y , z ) lies on the surface of x 2 y 2 r 2 . If we denote the angle between the projection vector
T
y
The Vector equation of the tangent to the curve Γ at P0 is

x
r ( t0 ) tr ( t0 )
5
The equation of the tangent line to curve Γ The Vector equation: r ( t0 ) tr ( t0 )
z
L
r0
a
r
or
r( t ) r 0 ta, t R,
where r ( x , y , z ) is the position vector
x
O
y
of the variable point P(x,y,z).
3
The Parametric Equations of a Space Curve
y y Γ1 Γ2
r ( t0 ) 0
2 : r( t ) ( t 3 , t 2 )
O
x
piecewise smooth curve
O x7
The normal plane to Γ
We have seen that for a given space curve Γ if r(t) is derivable at t0 and r′(t0) ≠ 0, then the tangent to Γ at P0 exists and is unique. There is an infinite number of straight lines through the point P0 , which are perpendicular to the tangent and lie in the same plane.
t1 t 2 , , then Γ is said to be a simple curve.
x
O
y
r
4
The tangent line to Γ
: r( t ) ( x( t ), y( t ), z ( t ))
z
r ( t 0 ) r ( t0 )
O P0
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 Γ at the corresponding point P0 . r′ (t0 ) is called the tangent vector to the curve Γ at P0 .
11
2. Tangent planes and normal lines of surfaces
z
Normal line
P0 ( x0 , y0 , z0 )
Tangent plane
O
y
x
12
Parametrizing
Any space point can be imagined that it lies on a sphere which is centered at the origin and the radius is x y z .
2 2 2
z

O
r
P ( xBiblioteka Baidu y, z )
If the angle between the projection vector
of OP on the xOy plane and the positive

x
y
direction of x-axis is denoted by θ, and
x r cos ,
y r sin , z z , 0 2 .
If P ( x , y, z ) is also a point of a space curve or a space surface, then we can parametrize the equation of the curve or surface.
of OP on the xOy plane and the positive
z
direction of x-axis is denoted by θ, and
r x y 0. Then the coordinate can
2 2
O
P ( x, y, z )
r
x
y
be expressed by
r ( t 0 ) 0
6
The Parametric equation:
The Symmetric equation:
The tangent line to Γ
A curve for which the direction of the tangent varies continuously is called a smooth curve. Example 1 : r( t ) (cos t ,sin t )
The plane is called the normal plane to the curve Γ at P0.
through the point P0 perpendicular to the tangent the equation of the normal plane
8
The normal plane to Γ
the angle between the vector OP and the positive direction of z-axis
is denoted by , then the two coordinate system are related by
x r sin cos ,
: r( t ) ( t ,2t 2 , t 2 ).
9
Tangent line and normal plane to a space curve
If the equations of the curve Γ is given in the general form F ( x , y , z ) 0, : G ( x , y , z ) 0, and the above equations of the curve Γ determine two implicit functions of one variable x, y=y(x) and z=z(x) in the neighbourhood U(P0) and both y(x) and z(x) have continuous derivative. Then
Similarly, a space curve Γ may also be represented by parametric equations or vector form
x x ( t ), y y( t ), z z ( t ), ( t ), r( t ) ( x( t ), y( t ), z ( t )) ( t ).
, a line in space can be expressed equations x x ( t ), y y( t ),( t ) by a parametric equations
x x0 lt , y y0 mt , t , z z nt , 0