北邮高等数学英文版课件Lecture-10-3

The Vector equation:
r(t0 ) tr(t0 )
The Parametric equation:
x(t ) x0 tx(t0 ),
y(t
)
y0
ty(t0 ),
z(t
)
z0
tz(t0
).
The Symmetric equation:
x x0 y y0 z z0 x(t0 ) y(t0 ) z(t0 )
y
Γ1
2 : r(t) (t3,t2)
y
Γ2
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.
Example Find the equations of the tangent line and the normal plane to the following curve Γ at point t=1.
r′ (t0 ) is called the tangent vector to the curve Γ at P0 .
O
The Vector equation of the
y tangent to the curve Γ at P0 is
x
r(t0 ) tr(t0 )
5
The equation of the tangent line to curve Γ
There is an infinite number of straight lines through the point P0 , which are perpendicular to the tangent and lie in the same plane.
The plane is called the normal plane to the curve Γ at P0.
equations
x x(t), y y(t ), z z(t ), ( t ),
or vector form
r(t) ( x(t), y(t), z(t)) ( t ).
If the vector valued function r(t) is continuous z
on the interval [ , ], then Γ is said to be a
equations x x(t), y y(t ),( t ), a line in space can be expressed
by a parametric equations
x x0 lt,
y
y0
mt,
t ,
z
z0
nt ,
or
r(t) r0 ta, t R,
z
The geometric meaning of the
: r(t) ( x(t), y(t), z(t))
z
P0
r(t0 )
r(twenku.baidu.com )
T
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 .
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
z
r O
P( x(t), y(t), z(t)) y
x
1) Tangent line and normal plane
x
y
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
L
r0
a
r y
O
where r ( x, y, z) is the position vector x of the variable point P(x,y,z).
3
The Parametric Equations of a Space
Curve
Similarly, a space curve Γ may also be represented by parametric
through the point P0 perpendicular to the tangent
the equation of the normal plane
8
The normal plane to Γ
The equation of the normal plane to the curve Γ at P0 is x(t0 )( x x(t0 )) y(t0 )( y y(t0 )) z(t0 )(z z(t0 )) 0
continuous curve;If Γ is a continuous curve and
y
O
and r(t1 ) r(t2 ) holds for any t1 , t2 ( , )
r
t1 t2 , , then Γ is said to be a simple curve. x
4
The tangent line to Γ
r(t0 ) 0
6
The tangent line to Γ
A curve for which the direction of the tangent varies continuously is called a smooth curve.
r(t0 ) 0
Example 1 : r(t) (cos t,sin t)