Differential geometrt
28 videos • 1,345 views • by msc mathematics
1
find level set and graph of a function f(X1,X2,...Xn+1)=X1²+X2²+...+(Xn+1)²
msc mathematics
Download
2
find (1) level set of a function(2) graph of a functionf:R²→R defined by f(X1,X2)=-X1²+X2²
msc mathematics
Download
3
show that graph of any function f:Rn→R is a level set for some function F:R(n+1)→R
msc mathematics
Download
4
Levi-Civita parallel vector field. ST if X & Y are Levi-Civita parallel ⇒ X. Y is constant along α
msc mathematics
Download
5
Define Geodesic. Show that geodesic have constant speed
msc mathematics
Download
6
sketch the vector field X on R² defined on X(p)=(p, x(p)) where X(p)=X(x1,x2)=(-x2,x1)
msc mathematics
Download
7
define orientation on the n surface. Give an example of an non orientable surface
msc mathematics
Download
8
Define parameterized n-surface in Rnk, k≥0
msc mathematics
Download
9
Find the integral curve through p=(1,1) of vector field on R² given by X(p)=(p, x(p)) & x(p)=(0,1)
msc mathematics
Download
10
Find the integral curve through p(1,0) & p(a, b) for the vector field X(p)=(x1,x2,-2x1,½x1) on U⊂R²
msc mathematics
Download
11
Find the integral curve through P(1,0) and P(a, b) for the vector field X(p)=(x1,x2,0,1)
msc mathematics
Download
12
Define parametrized n-surface
msc mathematics
Download
13
Find the normal curvatureof -x1²+x2²+x3²=1 at a point on the surface in the direction of V
msc mathematics
Download
14
Define Guass map. illustrate with an example
msc mathematics
Download
15
sketch the following vector field on R²X(p)=(p, x(p))where X(x1,x2)=(x2,x1)
msc mathematics
Download
16
sketch the level set of the function f(x1,x2,x3)=x1²+x2²+x3²at heights 0 and 4
msc mathematics
Download
17
Define (¡) differential 1 form and (¡¡) exact 1 form
msc mathematics
Download
18
describe the spherical image f(x1,x2,..,xn)=x1²+x2²+...+x(n+1)²
msc mathematics
Download
19
Find and sketch the gradient field of the function f(x1,x2)=x1+x2
msc mathematics
Download
20
define level set andgraph of a function.sketch it for the function f(x1,x2)=x1²-x2²
msc mathematics
Download
21
s be an n-surface in Rn+1.PT ∃ an I containing 0 and a geodesic α:I→S st α(0)=p & α •(0)=v and β:I→S
msc mathematics
Download
22
Δf(p)⊥ = α •(t)
msc mathematics
Download
23
prove that the gradient of f at p ∈ f-¹(c) is orthogonal to f-¹(c) at p
msc mathematics
Download
24
let c be an oriented plane curve. then ∃ a global parameterization of C iff C is connected
msc mathematics
Download
25
state and prove inverse function theorem for n surfaces
msc mathematics
Download
26
prove that the weingarten map is self adjoint
msc mathematics
Download
27
why weingarten map is called sphere operator
msc mathematics
Download
28
Evaluate the weingarten map Lp for X2²+X3²=a² in R³
msc mathematics
Download