1.1 (a) show that α(t) = (sin3t cost , sin3t sint, 0). (b). Find the equation of the tangent line to α at t = π/3. Elements Of Differentail Geometry by Richard s. Millman, George D. Parker

Elements Of Differentail Geometry  by Richard s. Millman, George D. Parker

Local Curve Theory

1.1 (a) show that α(t) = (sin3t cost , sin3t sint, 0)

Sol. 

    α(t) = (sin3t cost , sin3t sint, 0)

  Dα/dt = ( Sin3tsint + (-3costCost) ,sin3t(-cost) + (-3cost)sint , 0)

Dα/dt = ( Sin3tsint - 3costCost , -sin3tcost - 3costsint , 0)

Let suppose t = 0

Dα/dt = ( Sin3(0)sin(0) - 3cos(0)Cos(0) , -sin3(0)cos(0) - 3cos(0)sin(0) , 0)

=(0, -3 , 0 ,0 ) Regular Curve Proved. 

(b). Find the equation of the tangent line to α at t = π/3.

Sol. 

α(t) = (sin3t cost , sin3t sint, 0)

  Dα/dt = Sin3tsint + (-3costCost) ,sin3t(-cost) + (-3cost)sint.

Dα/dt = ( Sin3tsint - 3costCost , -sin3tcost - 3costsint , 0 )

.•.  Dα/dt = V

.•.  T = V / |v|

t = π/3.

V = ( Sin3(π/3)sin(π/3)  - 3cos(π/3) Cos(π/3) , -sin3(π/3)cos(π/3) - 3cos(π/3)sin(π/3) , 0 )

V =  ( 0 - 3/2 - 0 - 3√3/2 )

V =  - 3 - 3√3 / 2

|v| = √(1)^2+(3)^2

|v| = √10

T =  V / |v|

T =   - 3 - 3√3 / 2 /  √10 Ans.