Chapter 3: Surfaces in R^3 Flashcards
DEF 3.1 open subset
A subset U of R^n is open if, whenever P is a point in U,
there is a positive number ε such that every point Q ∈ R^n within a distance ε of P is also in U
- Q∈ R^n and ||PQ|| < ε implies Q∈U ( for p∈U)
- R^n and ∅ are trivial e.g any ε , e.g. no points in the set so no points to worry about
open subset examples
- R^n and ∅ are trivial e.g any ε , e.g. no points in the set so no points to worry about
- Products of open intervals : ]α_1,β_1 [ x…x]α_n, β_n[ for α_i less than β_i for i=1,…,n
eg n=1 [α, β] on real line
eg n=2 rectangle not including boundary ( can always find epsilon for any point)
eg n=3 cuboid without boundary
- The open ball eg B_R(P) = {Q∈R^n : ||PQ|| less than R} for P∈ R^n and R bigger than 0
disc without boundary
EXAMPLE: [α, β] is it open
Take [α, β] ⊂ R.
{x∈R: α ≤x≤ β}. If we take p=β then for any ε>0 we will have points not within the set.
e.g. take Q= p + 0.5ε. Then |QP| = 0.5ε less than ε. But Q= β. So Q not in [α, β]. This means [α, β] isn’t open
DEF: continuous at P in X
F is continuous at P ∈ X if, given any number ε bigger than 0, there is a number δ bigger than 0 such that
Q ∈ X and ||PQ|| < δ ⇒
||F(P)F(Q)|| < ε.
DEF continuous map F
The map F is continuous if it is continuous at every point in X
For subsets X ⊂R^m , Y⊂ R^n map F:X → Y is continuous at p ∈X if points near p are mapped by F to points in Y near F(P)
differentiability and continuity
all differentiable functions defined on an open set are continuous
DEF: 3.2.1
The partial derivative of a vector-valued function
σ : R^2 → R^3
σ(u, v) = (x(u, v), y(u, v), z(u, v))
Consider a vector-valued function σ : R^2 → R^3
σ(u, v) = (x(u, v), y(u, v), z(u, v))
The partial derivative of σ(u, v) with respect to u/ u-derivative,
is
∂σ/∂u= (∂x/∂u, ∂y/∂u, ∂z/∂u)
or σ_u(u, v) = (x_u(u, v), y_u(u, v), z_u(u, v))
which is another vector-valued function of the same form.
similarly wrt v
component by component
2nd order partial derivatives of a vector-valued function
σ : R^2 → R^3
σ(u, v) = (x(u, v), y(u, v), z(u, v))
∂^2σ/∂u^2,
∂^2σ/∂u∂v,
∂^2σ/∂v∂u,
∂^2σ/∂v^2,
or σuu, σuv, σvu, σvv.
usually functs..
In practice, we will always deal with functions σ : U → R^3
(where U ∈ R^2
is open) that are differentiable in both variables infinitely many times. In this case, σuv = σvu.
DEF: 3.2.3 dot product
Given two vector-valued functions σ_1, σ_2 : R^2 → R^3
σ_1(u, v) = (x_1(u, v), y_1(u, v), z_1(u, v))
σ_2(u, v) = (x_2(u, v), y_2(u, v), z_2(u, v))
their dot product σ_1 · σ_2 is the scalar-valued function (σ_1 · σ_2)(u, v) = x_1(u, v)x_2(u, v) + y_1(u, v)y_2(u, v) + z_1(u, v)z_2(u, v)
in R^3 dot product encodes lengths of vectors and angles for tangent vectors of surfaces.
PROP 3.2.4
a product rule
The partial derivatives of the dot product satisfy a product
rule:
(σ1 · σ2)_u = (σ1)_u · σ2 + σ1 · (σ2)_u;
(σ1 · σ2)_v = (σ1)v · σ2 + σ1 · (σ2)_v.
DEF 3.2.5: cross product
the cross product of σ_1(u, v) and σ_2(u, v) is the vector-valued function σ_1 × σ_2 given by (σ1 × σ2)(u, v) = (y1(u, v)z2(u, v) − z1(u, v)y2(u, v), z1(u, v)x2(u, v) − x1(u, v)z2(u, v), x1(u, v)y2(u, v) − y1(u, v)x2(u, v))
= det {
[ i j k ]
[x_1 y_1 z_1]
[x_2 y_2 z_2]
}
Example:
u derivative of dot product of σ(u, v) · σ(u, v)
Given a vector-valued function σ(u, v), the u-derivative of
σ(u, v) · σ(u, v) is
(σ · σ)u = σu · σ + σ · σu
= 2 σ_u · σ
and similarly its v-derivative is 2 σv · σ.
||σ|| constant
||σ|| constant⇒
σ_u(u, v) ⊥ σ(u, v),
σ_v(u, v) ⊥ σ(u, v)
for all (u, v) ∈ R^2.
In words, if a vector-valued function has constant magnitude, it is always orthogonal to its partial derivatives.
Definition 3.3.1. A parametrized surface in R^3
A parametrized surface in R^3
consists of an open set U ⊆
R^2 and a continuous map
σ : U → R^3
U is some open set on R^2 such as a product of open intervals
DEF:
surface
parametrization of S
The image S = σ(U) is called a surface and σ is called a parametrization
of S.
applying σ to open set gives some surface, image in R^3
EXAMPLE: For the plane P in R^3 that contains the point (1, 0, 0) and is parallel to the vectors (−1, 1, 0) and (−1, 0, 1) a parametrization is
a parametrization is
σ_1 : R^2 → R^3
(u, v) → (1 − u − v, u, v).
image is {(x,y,z) : x+y+z = 1}
EXAMPLE: For the cylinder C := {(x, y, z) | x^2 + y^2 = 1}
a parametrization is
a parametrization is
σ_2 : ]−10, 10[ × R → R^3
(u, v) → (cos(u), sin(u), v).
centred at origin, surface of revolution of a curve parametrized by γ: R → R^3, γ(t)= (1,0,t)
or by (θ,t) → (cos(θ), sin(θ), t)
EXAMPLE: For the paraboloid {(x, y, z) | z = x^2 + y^2} a parametrization is
a parametrization is
σ_3 : R^2 → R^3
(u, v) → (u, v, u^2 + v^2)
2d to 3d cup
EXAMPLE: For a function f : R^2 → R, the graph of f is the set Γ_f := {(x, y, z) | z − f(x, y) = 0}; a parametrization is
a parametrization is
σ_4 : R^2 → R^3
(u, v) → (u, v, f(u, v)).
EXAMPLE:
Find a parametrization of the plane P through the three
points (0, 0, 1), (0, 2, 3) and (1, 0, 0).
Any point on plane reached to from c by linear combo of vectors CA and CB
ie if x ∈ P then x=(1,0,0) + u(-1,0,1) +v(-1,2,3) for u,v ∈ R. So we can parametrize p by σ:R^2 → R^3 σ(u,v) =(1-u-v, 2v, u+3v)
parametrized surface of revolution
Let γ: ]α, β[ → R^3, t → (x(t), 0, z(t)) be a regular parametrization of a curve C in the xz-plane. The parametrized surface of revolution of C is
σ : R^2 → R^3, (θ, t) →(x(t) cos θ, x(t) sin θ, z(t))
curve is in plane x-z and revolves about z axis
LATITUDES
The circles of constant t are called latitudes
LONGITUDES
the lines of constant θ are
called longitude
