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find the curvature at (0,a) of the curve (x^2+y^2)^2 =a^2(y^2-x^2)
find the curvature and torsion of the curve z=u,y=1+u/u,z=1-u^2/ufind the curvature and torsion of the curve z=u,y=1+u/u,z=1-u^2/u
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
