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Differential Geometry | Topology
Prove that the equation normal to the rectangular hyperbola xy = c^2 at the point P(ct, c/t) is t^3*x - ty = c(t^4 - 1)
Differential Geometry | Topology
A particle P moves on the curve with polar equation r = 1/ (2 - sinx) . Given that at any instant t, during the motion, r^2 (dx/dt) = 4,
(i) write an expression for r(dx/dt) in terms of x.
(ii) Show that dr/dt = 4cosx and 1/3 <=r<=1.
(iii) Find the speed of P when x = 0.
(iv) Prove that the force acting on P is directed towards the pole.
(i) write an expression for r(dx/dt) in terms of x.
(ii) Show that dr/dt = 4cosx and 1/3 <=r<=1.
(iii) Find the speed of P when x = 0.
(iv) Prove that the force acting on P is directed towards the pole.
Differential Geometry | Topology
Find the equation of the tangent to the plane f(x,y) at the point (-1,2)
F(X,y )= 2x^2 -e^2x-3y-8
Differential Geometry | Topology
Shade the region of the xy-plane for which
a) (𝑥2+𝑦2−16)(𝑥2−4)≤0
b) (𝑦−𝑥)(𝑦2+𝑥3)>0
Differential Geometry | Topology
Find the rectangular coordinates of the point whose spherical coordinates are
1) (5, π/2,π/2)
2) (4, π/3, 2π/3)
3) (0,π/11,π/5)
4) (2, 5π/3, 3π/4)
1) (5, π/2,π/2)
2) (4, π/3, 2π/3)
3) (0,π/11,π/5)
4) (2, 5π/3, 3π/4)
Differential Geometry | Topology
Find the spherical coordinates of the point whose coordinates are
1) (1, 1, √6)
2) (-2, 2√3, 4)
3) (-√3, 1, -2)
4) (4,-4√3, 6)
1) (1, 1, √6)
2) (-2, 2√3, 4)
3) (-√3, 1, -2)
4) (4,-4√3, 6)
Differential Geometry | Topology
Change the following from cylindrical coordinates to rectangular coordinates
1) (5, π/6, 3)
2) (6, π/3, -5)
1) (5, π/6, 3)
2) (6, π/3, -5)
Differential Geometry | Topology
Show that. The curvature and torsion of the straight line is zero.
Differential Geometry | Topology
Calculate the normal and the geodesic curvatures of the following curves on
the given surfaces:
(b) The right circular helix γ(θ) = (a cos θ, a sin θ, bθ) on the right circular cylinder
σ(u, v) = (a cos u, a sin u, v), where a, b > 0 are constants.
the given surfaces:
(b) The right circular helix γ(θ) = (a cos θ, a sin θ, bθ) on the right circular cylinder
σ(u, v) = (a cos u, a sin u, v), where a, b > 0 are constants.
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.
Show that FIII = FIIF^−1I FII , where FI and FII are the 2 × 2 symmetric matrices
associated with the first and the second fundamental forms, respectively
||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.
Show that FIII = FIIF^−1I FII , where FI and FII are the 2 × 2 symmetric matrices
associated with the first and the second fundamental forms, respectively
