High number_ Chapter 2 differential calculus of multivariate functions__ Geometric application of partial derivatives_ Tangent and normal plane of space curve

One Definition

Definition ： Curves in a given space L,    spot P₀ yes L A fixed point on , set sth. up P yes L Above different from P₀ The point of ,  A straight line P₀P be called after P₀ A secant of .  When P Along the curve L Infinite approach point P₀ when ,  The limit position of secant P₀T It is called a curve L At point P₀ Of Tangent line ,    Passing point P₀ And perpendicular to the tangent plane  π It is called curve at point P₀ Of The normal plane .

Given space curve L

L The points on the P₀( x₀,  y₀,  z₀) The corresponding parameter is t₀,  namely x₀ = x(t₀),  y₀= y( t₀),   z₀ = z(t₀).

When secant tends to tangent , The direction vector of secant tends to the direction vector of tangent , Therefore, the direction vector of the tangent can be taken as

s = {x'(t₀),     y'(t₀),   z'(t₀)},    So tangent P₀T The equation is

Because the direction vector of the tangent is the normal plane  π The normal vector of ,  Therefore, according to the point normal equation of the plane, we can get the normal plane  π The equation is

We're going to vector     It is called a curve L At point P₀ Tangent vector of .

Two   See the sample