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Find the equation of the tangent plane of the following surface patches
at the indicated points:
(b) σ(r, θ) = (r cos θ, r sin θ, 2θ), P =

3, 1,
π
3
Find a countable dense subset of (C[0,1],ρ) where ρ denotes the uni-form metric.(You do not need to justify)
If f:R^n-->R is a smooth function, define hypersurface f(x)=c for some constant c, then prove velocity vector is orthogonal to the normal vector i.e <\nabla f(p),v_{\gamma}(p)>=0
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


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