Solutions To Differential Geometry Problems (Part 3)

 1) Sketch the graphs of;

x ₂  = - Ö ( 8 ²   -   x ₁ ²²)

And:

x ₂  = - Ö ( 16 ²   -   x ₁ ²²)

On the same Cartesian axes.

Soln.

2)(a) Write the polar form of the equation of the line:

3 x ₁   +   4 x ₂   =  5

Soln.

x ₁ = r cos q

x ₂   =  r sin q

then:  3 x ₁   +   4 x ₂   =  3 r cos q + 4 r sin q

The polar form of the equation is:

r [3 cos q + 4 sin q]  =  5

b)Determine the polar (r,  q) equation for :

x ₁ ²  + x ₂ ²   -  2ax ₂  =  0,    a  ≠   0

And sketch the resulting curve.

Soln.

x ₁ ²    +    x ₂ ²  =   ( r cos q ) ²  +  ( r sin q ) ²

=   r ² (cos ² q   +   sin ² q) 

=> (eqn. of circle)   r    =   2 a x ₂  =   2a  sin q  

3)(a)  Let r and  q  be polar coordinates in the x ₁ x ₂ - plane. Give the representation of the following curve in Cartesian coordinates:

r =   a q

And sketch it.

Soln.

 (x ₂ -   x ₁)  tan  Ö ( x ₁ ²  +   x ₂ ²²) /  a  =    0

This is an Archimedian spiral:

b)  A student's analysis of the curve (cardioid):

 r =   6 (1 - cos q),    is shown below:

Using differential calculus show how an expression for the angle y  can be obtained in terms of the angle   q.

Soln.

r =   6 (1 - cos q)

dr =   6 sin q  dq

tan y   =  r dq /  dr

tan y   =  6 (1 - cos q) d q /  6 sin q  dq

=   tan q/ 2