Begin2.DVI

Analysis of this equation demonstrates that the velocity vector
v is directed along

the tangent vector at any time t and has the magnitude given by v = dsdt which

represents the speed of the particle.

The derivative of the velocity vector with respect to time tis the acceleration

and

a =d
v

dt

=ˆetdv

dt

+d

ˆet

dt

v.

From the Frenet-Serret formula and using chain rule differentiation, it can be shown

that the time rate of change of the unit tangent vector is

dˆet

dt

=dˆet

ds

ds

dt

=v

ρ

ˆen.

Substituting this result into the acceleration vector gives

a =dv

dt

ˆet+v

2

ρ

ˆen.

The resulting acceleration vector lies in the osculating plane. The tangential compo-

nent of the acceleration is given by dvdt ,and the normal component of the acceleration

is given by v

2

ρ.

Surfaces

A surface can be defined

(i) Explicitly z=f(x, y )

(ii) Implicitly F(x, y, z ) = 0

(iii) Parametrically x=x(u, v ), y =y(u, v ), z =z(u, v )

(iv) As a vector
r =
r (u, v ) = x(u, v )ˆe 1 +y(u, v )ˆe 2 +z(u, v )ˆe 3

or
r =
r (x, y ) = xˆe 1 +yˆe 2 +f(x, y )ˆe 3

(v) By rotating a curve about a line.

Here again it should be noted that the parametric representation of a surface is not

unique.

If the functions used to define the above surfaces are continuous and differ-

entiable functions and are such that the functions defining the surface and their

partial derivatives are all well defined at points on the surface, then the surfaces

are called smooth surfaces. If the surface is defined implicitly by an equation of the

form F(x, y, z ) = 0, then those points on the surface where at least one of the partial

derivatives ∂F∂x , ∂F∂y , ∂F∂z is different from zero are called regular points on the surface.

If all of these partial derivatives are zero at a point on the surface, then that point

is called a singular point of the surface.