When a Riemannian metric is of the form ds? = (U + V) (dudu + dvdv)....
Find the first quadratic form of a surface called helicoid, r(u, v)-(u cos v, u sin v, av), where a is a constant parameter. Then find the angle of intersection of lines on the surface of helicoid given by equations u v0, u0.
Find the first quadratic form of a surface called helicoid, r(u, v)-(u cos v, u sin v, av), where a is a constant parameter. Then find the angle of intersection of lines on the surface of helicoid...
Let R^2 be equipped by the metric ds^2 = (4/(1 + x^2 + y^2 )^2)
(dx^2 + dy^2 ), i.e. its first fundamental form is E = G =
4/(1+x^2+y^2)^2 , F = 0. Use the formula dω12 = −Kω1 ∧ ω2 to
calculate its Gauss curvature.
Let R2 be equipped by the metric i.e. its first fundamental form is E = G = TrtFF, F = 0. Use the formula dui,-- . КМ Л w2 to calculate its Gauss...
Differential Geometry
Prove that for a coordinate patch x(u,v), where U is the unit
normal defined as
, and K is the Gaussian Curvature.
L, V 1,0) (0,1 1,0) We were unable to transcribe this image
10. Evaluate the integral le z ds, where S is given by r(u, v) = (u + v)i + (u – v)j + sin v k, Osus2, 0505
4. Consider the surface of revolution o(u, v) (f(u)cosv, f(u) sin v, g(u)) where uf(u), 0, g(u)) is the unit-speed regular curve in R3, Find the normal curvature of meridian v constant and geodesic curvatures of a parallel u=constant.
4. Consider the surface of revolution o(u, v) (f(u)cosv, f(u) sin v, g(u)) where uf(u), 0, g(u)) is the unit-speed regular curve in R3, Find the normal curvature of meridian v constant and geodesic curvatures of a parallel u=constant.
Part C
The standard Euclidean metric on R3 is simply ds? = d.x2 + dy? + dz2. (a) Explain why I c = 0) in this basis. (b) Now consider the same metric expressed in spherical coordinates, ds2 = dr2 + pºde2 + sin Odo2). Determine the Christoffel symbols in this coordinate system using the same method as in the earlier question. (c) Write down Laplace's equation v2f = gabv. Vof = 0) where f = f(r,0,0). What happens if...
Find the component form of u + v given the lengths of u and V and the angles that u and v make with the positive x-axis. Tull = 7, y = 0 Ilvl = 2, By = 60° u + V
1. Let (X, d) be a metric space, and U, V, W CX subsets of X. (a) (i) Define what it means for U to be open. (ii) Define what it means for V to be closed. (iii) Define what it means for W to be compact. (b) Prove that in a metric space a compact subset is closed.
Evaluate the surface integral. y ds, S is the helicoid with vector equation r(u, v) = (u cos(V), u sin(), v), OSUS 4,0 SV S.
. (a) Show that the function u= 4x2 - 12.xy2 is harmonic and v=12.xy-4v2 is a harmonic conjugate of u. [Consequently, the function f =u+iv is entire, thus it has an antiderivative and that any contour integral of f is path independent.] (b) Find an antiderivative F(-)= F(x+iy)=P(x, y)+i Q(x, y) of the function f; and (c) evaluate ( f (2) ds , where C is any contour from 0 to 1–2i .