Differential Geometry | Topology Answers

The position vector r of a moving particle at time t after the start of the motion is given by

r =5(1+4t)i+5(19+2t−t2)j.

a) Find the initital velocity of the particle.

b) At time t = T, the particle is moving at right angles to its initial direction of motion. Find the value of T and the distance of the particle from its initial position at this time.

Find the curvature and torsion of the curve x=a(u-sinu),y=(1-cosu),z=bu

let f: x→ y and g: y →z be continuous. show that gof: x→ z is a homeomorphism then g one to one implies that f and g are homeomorphism

Calculate the first fundamental forms of the following surfaces:

i. 𝝈(𝑢,𝑣)= (sinh𝑢sinh𝑣,sinh𝑢cosh𝑣,sinh𝑢)

ii. 𝝈(𝑢,𝑣) = (𝑢 − 𝑣,𝑢 + 𝑣, 𝑢^2+ 𝑣^2).

A plane curve is given by γ(θ)=(rcosθ,rsinθ)

γ(θ)=(rcos⁡θ,rsin⁡θ), where r is a smooth function of θ

θ (so that (r,θ)

(r,θ) are the polar coordinates of γ(θ)

γ(θ)). Under what conditions is γ

γ regular? Find all functions r(θ)

r(θ) for which γ

γ is unit-speed. Show that, if γ

γ is unit-speed, the image of γ

γ is a circle; what is its radius?

At any point of the path x=3cos⁡t,y=3sin⁡t,z=4t, what is the Normal vector?

The principal value of the argument of

(1+i√3)(1−i) is____

Suppose that \\( \\alpha=2i-3j+k\\) and \\(\\beta=7i-5j+k\\) , find a unit vector perpendicular to \\(\\alpha\\) and \\(\\beta) respectively.

Find the first fundamental form of surface of revolution?

If G⃗=〖5t〗2i+tj−t3k and

F⃗=sinti–costj, what is d/dt(G×F⃗) ?