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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
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

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

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


Sketch the gradient field ∇f of the following functions:
(a) f(x ,y) = x^2 + y^2.
Sketch the level curves f
1(c) for the following functions:
f(x, y, z) = x-y2-z2
, c = 1, 0, 1

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.


Prove them

K=|r'×r"|/|r'|^3


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


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.
Sketch the gradient field ∇f of the following functions:
f(x , y) = (x^2 −y^2)/4
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.
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