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Find the curvature, the radius and the center of curvature at a point.


r=1+ cos theta ,theta=π/2



Find killing equations for spherical polar coordinates


DG. Let X be the universal space. Which set is equal to "\\mathrm{Cl}(X) \\cap \\mathrm{Cl}(X \\backslash A) ?(\\mathrm{Fr}" stands for the boundary of a set, "\\mathrm{Cl}" means the closure of a set).


"\\text { Find the interior Int }\\left\\{\\frac{1}{n} \\mid n \\in \\mathbb{N}\\right\\} \\text { of a subset of } \\mathbb{R} \\text {. }" (DG)


"\\text { Find the limit points of }\\left\\{\\frac{1}{n} \\mid n \\in \\mathbb{N}\\right\\} \\text { in } \\mathbb{R} \\text {. }" (DG)


Find the evolute of

x

2

a2

y

2

b

2

= 1 as the envelope of the normals.




4. Find the evolute of the rectangular hyperbola y


2 = 4ax. Ans: 27ay2 = 4(x −


2a)

3

Hint: Take P (at2


, 2at) be any point on the parabola.

5. Find the envelope of the family of lines of the form y = mx±

√︀

a2m2 − b

2. Ans:


x

2

a2

y

2

b

2

= 1


6. Find the envelope of the family of lines of the form

x

a

+

y

b

= 1 subject to the


condition a + b = 1. Ans: √

x +

b = 1


7. Find the evolute of

x

2

a2

+

y

2

b

2

= 1 as the envelope of the normals. Ans: (ax)

2/3 +


(by)

2/3 = (a

2 − b

2

)

2/3



1. Find the radius of curvature (ROC) at (a cos3 θ, a sin3


θ) on the curve x

2/3 +


y

2/3 = a

2/3



2. Find the radius of curvature (ROC) for the curve x = a(cos θ + θ sin θ), y =

a(sin θ − θ cos θ).


find the radius of the curvature at (acostheta^3,asintheta^3) on the curve x^2/3+y^2/3=a^2/3


Find




the Curvature and




torsion of the Curves r = (u, (1 + u)/u, (1 - u ^ 2)/u)

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