Differential Geometry | Topology

Show that the Mercator projection

σ(u ,v) = (sech u cos v, sech u sin v ,tanh u)

is a regular surface patch of the unit sphere

σ(u ,v) = (sech u cos v, sech u sin v ,tanh u)

is a regular surface patch of the unit sphere

Differential Geometry | Topology

Show that the Mercator projection

σ(u, v) = (sech u cos v,sech u sin v,tanh u)

is a regular surface patch of the unit sphere

σ(u, v) = (sech u cos v,sech u sin v,tanh u)

is a regular surface patch of the unit sphere

Differential Geometry | Topology

Sketch the level curves f−1(c) for the following functions:

f(x, y, z) = x − y2 − z2, c = −1, 0, 1.

Differential Geometry | Topology

Sketch the gradient ﬁeld ∇f of the following functions:

(a) f(x ,y) = x^2 + y^2.

(a) f(x ,y) = x^2 + y^2.

Differential Geometry | Topology

Sketch the level curves f

1(c) for the following functions:

f(x, y, z) = x-y2-z2

, c = 1, 0, 1

1(c) for the following functions:

f(x, y, z) = x-y2-z2

, c = 1, 0, 1

Differential Geometry | Topology

Sketch the level curves f−1(c) for the following functions:

(a) f(x, y) = x2 + y2, c = 0, 1, 2, 3, 4.

(b) f(x, y, z) = x − y2 − z2, c = −1, 0.

Differential Geometry | Topology

Prove them

K=|r'×r"|/|r'|^3

T=[r', r",r"]/K^2(r')^6

Differential Geometry | Topology

Sketch the level curves f−1(c) for the following functions:

a) f(x ,y ,z) = x−y^2 −z^2, c = −1,0,1.

a) f(x ,y ,z) = x−y^2 −z^2, c = −1,0,1.

Differential Geometry | Topology

Sketch the gradient ﬁeld ∇f of the following functions:

f(x , y) = (x^2 −y^2)/4

f(x , y) = (x^2 −y^2)/4

Differential Geometry | Topology

Show that the circular cylinder S = {(x, y, z) ∈R^3 : y^2+z^2 = 1}can be covered by a single regular surface patch, and hence is a surface.