Differential Geometry | Topology Answers

1. Verify that (xy² -1) dx + (x²y) dy is exact and hence solve it.
2. Solve: dy/dx - y = e^2x , y(0) = 7
3. The differential equation of motion of a simple pendulum is given as d²x/dt² + g/l*x where x is the dependent variable and t is the independent variable. Find the displacement of the pendulum as a function of time t.  
4. Solve the equation y^//+ y^/- 2y- x² = 0 

The fourth-degree polynomial

f(x) = 230x⁴ + 18x³ + 9x− 221x − 9

has two real zeros, one in [−1, 0] and the other in [0, 1]. Attempt to approximate these zeros to within

10^-2 using the

(a)

Secant method(Use the endpoints of each interval as the initial approximations),

(b)

Newtons method(Use the midpoints of each interval as the initial approximation).

Let M and N be smooth manifolds and let πM : M × N → M and πN : M × N → N

be the projection maps. For any (p, q) ∈ M × N show that the map

Π : Tp(M × N) → TpM × TpN,

defined by

Π(v) = (D(πM)p(v), D(πN )q(v))

is an isomorphism

1-(Invariance of dimension) Let M and N be smooth manifolds and suppose M is diffeo-

morphic to N. Then show that dim M = dim N.

2-(Inverse function theorem) Let M and N be smooth manifolds and let F : M → N be

smooth. Suppose DFp : TpM → TF(p)N is an isomorphism for each p . Then show that

M is locally diffeomorphic to N.

3-Let M and N be smooth manifolds and let πM : M × N → M and πN : M × N → N

be the projection maps. For any (p, q) ∈ M × N show that the map

Π : Tp(M × N) → TpM × TpN,

defined by

Π(v) = (D(πM)p(v), D(πN )q(v))

is an isomorphism.

Let X={1,2,3,4,5,6}.Then give two examples to show that every T1 space is t0 but Converse is not true.

find the curvature at (0,a) of the curve (x^2+y^2)^2 =a^2(y^2-x^2)