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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.
Differential Geometry | Topology
What is moving trihedron?
Differential Geometry | Topology
Sketch the gradient field ∇f of the following functions:
(a) f(x,y) = x^2 + y^2.
(b) f(x,y) = x^2 −y^2/4 .
(a) f(x,y) = x^2 + y^2.
(b) f(x,y) = x^2 −y^2/4 .
Differential Geometry | Topology
Find the equation of the tangent plane of the following surface patches at the indicated points: (a) σ(u,v) = (u,v,u2 + v2), P = (1,√2,3).
(b) σ(r,θ) = (rcosθ,rsinθ,2θ), P =(√3,1, π 3).
(b) σ(r,θ) = (rcosθ,rsinθ,2θ), P =(√3,1, π 3).
Differential Geometry | Topology
Let (a,b,c ) not equal to 0 be real constants. Show that the ellipsoid
E =((x,y,z) ∈R^3 : x^2/a^2 + y^2/b^2 + z^2/c^2 = 1) is a smooth surface.
E =((x,y,z) ∈R^3 : x^2/a^2 + y^2/b^2 + z^2/c^2 = 1) is a smooth surface.
Differential Geometry | Topology
(a) 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.
(b) Draw a picture describing this surface patch.
(b) Draw a picture describing this surface patch.
Differential Geometry | Topology
(a) Show that the circular cylinder S = {(x,y,z) ∈R3 : y2+z2 = 1} can be covered by a single regular surface patch, and hence is a surface.
(b) Draw a picture describing this surface patch.
(b) Draw a picture describing this surface patch.
Differential Geometry | Topology
Discuss the formula for "K" and "T" when positive vector is in the form of general parameter of "u".
K=|r'×r"|/|r'|^3
T=[r', r",r"]/K^2(r')^6
K=|r'×r"|/|r'|^3
T=[r', r",r"]/K^2(r')^6
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