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Differential Geometry | Topology
a) Calculate the principal, Gaussian, and mean curvatures of the surface of revolution
σ(u, v) = (f(u) cos v, f(u) sin v, g(u)), where f, g are smooth real-valued functions
with f > 0.
b) Hence find these curvatures in the following cases:
i. f(u) = e^u, g(u) = u.
ii. f(u) = 2 + sin u, g(u) = u.
σ(u, v) = (f(u) cos v, f(u) sin v, g(u)), where f, g are smooth real-valued functions
with f > 0.
b) Hence find these curvatures in the following cases:
i. f(u) = e^u, g(u) = u.
ii. f(u) = 2 + sin u, g(u) = u.
Differential Geometry | Topology
Calculate the normal and the geodesic curvatures of the following curves on
the given surfaces:
(a) The circle γ(t) = (cost, sin t, 1) on the paraboloid σ(u, v) = (u, v, u^2 + v^2).
the given surfaces:
(a) The circle γ(t) = (cost, sin t, 1) on the paraboloid σ(u, v) = (u, v, u^2 + v^2).
Differential Geometry | Topology
Calculate the first fundamental form, the second fundamental form, and the
Weingarten matrix of the following surface patches of the unit sphere:
(a) σ(θ, φ) = (cos θ cos φ, cos θ sin φ,sin θ).
(b) σ(u, v) = (sech u cos v,sech u sin v,tanh u).
Differential Geometry | Topology
The third fundamental form of a surface σ(u, v) is
||N̂u|| ^2 du^2 + 2N̂u.N̂v dudv + ||N̂v||^2 dv^2
where N̂ (u, v) is the standard unit normal to σ(u, v). Let FIII be the associated 2 × 2
symmetric matrix.
a) ) Show that FIII − 2HFII + KFI = 0, where K and H are the Gaussian and mean
curvatures, respectively, of σ.
||N̂u|| ^2 du^2 + 2N̂u.N̂v dudv + ||N̂v||^2 dv^2
where N̂ (u, v) is the standard unit normal to σ(u, v). Let FIII be the associated 2 × 2
symmetric matrix.
a) ) Show that FIII − 2HFII + KFI = 0, where K and H are the Gaussian and mean
curvatures, respectively, of σ.
Differential Geometry | Topology
Find the evolute of the four cusped hypocycloid
x=a cos^3θ y=a sin^3θ
x=a cos^3θ y=a sin^3θ
Differential Geometry | Topology
Prove that for cardiod r=a(1+cosθ)
ρ^2/r is constant
ρ^2/r is constant
Differential Geometry | Topology
For curve r^m=a^m cosmθ Prove that
P=a^m÷(m+1)r^m-1
P=a^m÷(m+1)r^m-1
Differential Geometry | Topology
Find the centre of curvature of astroid
x^2/3+y^2/3=a^2/3
x^2/3+y^2/3=a^2/3
Differential Geometry | Topology
Find the centre of curvature of the curve
x=3t y=t^2-6 at (x,y)
x=3t y=t^2-6 at (x,y)
Differential Geometry | Topology
Find envelope of the family of straight line
y=mx+a/m
y=mx+a/m
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