Differential Geometry | Topology Answers

If G⃗=〖5t〗2i+tj−t3k and

F⃗=sinti–costj, what is d/dt(G×F⃗) ?

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At any point of the path x=3cos⁡t,y=3sin⁡t,z=4t, find the Tangent vector . n

Consider the cylinder S : y² + z² = 1.

(a) write down the surface patch σ for the cylinder S.

(b) Write down the geodesic equations for σ.

(c) Find two different geodesics on S. Justify your answer.

Consider the cylinder S : y² + z² = 1.

(a) write down the surface patch σ for the cylinder S.

(b) Write down the geodesic equations for σ.

(c) Find two different geodesics on S. Justify your answer.

Consider a unit surface S = {(x, y, x) : x2 + y2 + z2 = 1} with a parametrization

S : σ(u, v) = (sin(u) cos(v),sin(u) sin(v), cos(u)).

Consider the cylinder S : y² + z² = 1.

(a) write down the surface patch σ for the cylinder S.

(b) Write down the geodesic equations for σ.

(c) Find two different geodesics on S. Justify your answer.

Consider a plane with the surface patch σ(u, v) = (1+2u+3v, u-v,-2+u-4v). Verify the Gauss equations for σ.

geodesic equations for surface patch of cylinder

Let S be an arc length parameterized curve in R³. Suppose kt not equal to zero at a point P of all the plane consisting the tangent line to S at P. Show that S lies locally on both side only of the osculating palne.