Chapter 9 Vectors and the Geometry of Space (向量與空間幾何)

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Chapter 9 Vectors and the Geometry of Space (向量與空間幾何)

Hung-Yuan Fan (范洪源)

Department of Mathematics, National Taiwan Normal University, Taiwan

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本章預定授課範圍

9.0 Definitions and Preliminaries 9.6 Surfaces in Space

9.7 Cylindrical and Spherical Coordinates

Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan Chapter 9, Calculus B 2/60

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Section 9.0

Definitions and Preliminaries

(定義和預備知識)

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Vectors in the Plane (平面向量)

Two-Dimensional Euclidean (Vector) Space R² ={(v1, v2)| v1, v2∈ R}

={v = ⟨v1, v₂⟩ | v is a vector (向量)}

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Definitions and Notations (1/2)

Vector Addition (向量加法):

v + u =⟨v1, v2⟩ + ⟨u1, u2⟩ = ⟨v1+ u1, v2+ u2⟩ ∀ v, u ∈ R². Scalar Multiplication (純量乘法):

cv = c⟨v1, v₂⟩ = ⟨cv1, cv₂⟩ ∀ c ∈ R and v ∈ R². Length or Norm (範數) of a Vector:

∥v∥ = ∥⟨v1, v₂⟩∥ =q

v²₁+ v²₂ ≥ 0 ∀ v ∈ R².

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Definitions and Notations (2/2)

If i =⟨1, 0⟩ and j = ⟨0, 1⟩ are standard unit vectors in R², then v =⟨v1, v2⟩ = v1i + v₂j ∀ v ∈ R².

If v is represented by the directed line segment from P(p₁, p₂) to Q(q₁, q2), then it has the component form (分量形式)

v =⟨v1, v₂⟩ = ⟨q1− p1, q₂− p2⟩ ∈ R².

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Vectors in Space (空間向量)

Three-Dimensional Euclidean (Vector) Space R³ ={(v1, v2, v3)| v1, v2, v3 ∈ R}

={v = ⟨v1, v₂, v₃⟩ | v is a vector in space}

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Vectors in Space

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Definitions in R

³

(1/2)

Vector Addition (向量加法):

v + u =⟨v1, v2, v3⟩ + ⟨u1, u2, u3⟩

=⟨v1+ u₁, v₂+ u₂, v₃+ u₃⟩ ∀ v, u ∈ R³. Scalar Multiplication (純量乘法):

cv = c⟨v1, v2, v3⟩ = ⟨cv1, cv2, cv3⟩ ∀ c ∈ R and v ∈ R³. Length or Norm of a Vector:

∥v∥ = ∥⟨v1, v2, v3⟩∥ =q

v²₁+ v²₂+ v²₃ ≥ 0 ∀ v ∈ R³.

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Definitions in R

³

(2/2)

If i =⟨1, 0, 0⟩, j = ⟨0, 1, 0⟩ and k = ⟨0, 0, 1⟩ are standard unit vectors in R³, then

v =⟨v1, v₂, v₃⟩ = v1i + v₂j + v₃k ∀ v ∈ R³.

If v is represented by the directed line segment from

P(p₁, p₂, p₃) to Q(q₁, q₂, q₃), then it has the component form v =⟨v1, v₂, v₃⟩ = ⟨q1− p1, q₂− p2, q₃− p3⟩ ∈ R³.

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向量 −→ PQ 的示意圖 (承上頁)

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The Dot Product of Vectors (向量內積)

Def. (向量內積的定義)

(1) The dot product of u =⟨u1, u2⟩ and v = ⟨v1, v2⟩ is u• v = u1v₁+ u2v₂ ∈ R.

(2) The dot product of u =⟨u1, u₂, u₃⟩ and v = ⟨v1, v₂, v₃⟩ is u• v = u1v₁+ u₂v₂+ u₃v₃ ∈ R.

(3) In some textbooks, the dot product is also called the inner product of vectors.

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Some Special Vectors

Let u and v be vectors inR² or R³.

∥v∥v is the unit vector in the direction of v̸= 0.

(沿著 v 方向的單位向量)

u and v are parallel vectors (平行向量) if ∃ c ∈ R s.t. u = cv. u and v are orthogonal vectors (垂直向量) if u• v = 0.

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Cross Product of Vectors in R

³

Def. (空間向量的外積)

The cross product of u =⟨u1, u₂, u₃⟩ and v = ⟨v1, v₂, v₃⟩ is

u× v =

i j k

u₁ u₂ u₃ v₁ v₂ v₃

(對第一列作行列式降階!)

=

u₂ u₃ v₂ v₃

i−

u₁ u₃ v₁ v₃

j +

u₁ u₂ v₁ v₂

k.

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Notes

u× v is a vector in R³, but u• v is a scalar.

(u× v) • u = 0 = (u × v) • v, i.e., the vector u× v is orthogonal to u and v, respectively.

v× u = −(u × v), i.e., they are parallel vectors, but in the opposite directions.

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外積的示意圖 (承上頁)

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Section 9.6 Surfaces in Space

(空間中的曲面)

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Type I: Cylindrical Surfaces (柱狀曲面或是柱面)

If the line L is not parallel to the plane containing a curve C, then S = {ℓ | ℓ is a line parallel to L and intersecting C}

is a cylindrical surface, or simply a cylinder.

C is the generating curveof S.

The lines parallel to L arerulings.

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柱面的示意圖 (承上頁)

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Example

The right circular cylinder (圓柱面) with radius a > 0 is defined by S = {(x, y, z) ∈ R³| x²+ y²= a²}.

Then the generating curve of the cylinderS lying in the xy-plane is C : x²+ y² = a².

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圓柱曲面的示意圖 (承上頁)

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Example 1 的示意圖

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Type II: Quadratic Surfaces (二次曲面)

The general equation of a quadratic surface is

Ax²+ By²+ Cz²+ Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0, where the coeﬀicients A, B,· · · , J are real numbers.

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Type II: Quadratic Surfaces

(1) Ellipsoid: (橢圓曲面)

x² a² +y²

b² +z²

c² = 1 with a, b, c > 0.

Them the xy-trace, xz-trace and yz-trace of the surface are x²

a² + y²

b² = 1, x² a² +z²

c² = 1 and y² b² +z²

c² = 1.

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Ellipsoid 的示意圖 (承上頁)

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Example 4 的示意圖

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Type II: Quadratic Surfaces

(2) Hyperboloid of One Sheet: (單片雙曲面) x²

a² +y² b² −z²

c² = 1 with a, b, c > 0.

a² + y²

b² = 1, x² a² −z²

c² = 1 and y² b² −z²

c² = 1.

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Hyperboloid of One Sheet 的示意圖

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Type II: Quadratic Surfaces

(3) Hyperboloid of Two Sheets: (雙片雙曲面) z²

c² − x² a² −y²

b² = 1 with a, b, c > 0.

Them the xz-trace and yz-trace of the surface are z²

c² − x²

a² = 1 and z² c² − y²

b² = 1, but no xy-trace!

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Hyperboloid of Two Sheets 的示意圖

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Example 2 的示意圖

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Type II: Quadratic Surfaces

(4) Elliptic Cone: (橢圓錐面)

x² a² + y²

b² − z²

c²= 0 with a, b, c > 0.

a² + y²

b² = k for some k≥ 0, x²

a² −z²

c² = 0 =⇒z =±c axand y²

b² −z²

c² = 0 =⇒z =±c by.

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Elliptic Cone 的示意圖

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Type II: Quadratic Surfaces

(5) Elliptic Paraboloid: (橢圓拋物面) z = x²

a² +y²

b² with a, b > 0.

a² + y²

b² = k for some k≥ 0, z = x²

a² and z = y² b².

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Elliptic Paraboloid 的示意圖

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Example 3 的示意圖

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Type II: Quadratic Surfaces

(6) Hyperbolic Paraboloid: (雙曲拋物面) z = y²

b² − x²

a² with a, b > 0.

Them the xy-trace, xz-trace and yz-trace of the surface are y =±b

ax, z =−x²

a² and z = y² b².

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Hyperbolic Paraboloid 的示意圖

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Section 9.7

Cylindrical and Spherical Coordinates (柱面坐標與球面坐標)

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Def. (柱面坐標系統)

In the cylindrical coordinate system, a point P(x, y, z)∈ R³ is represented by an ordered triple (r, θ, z) with

(r, θ) is the polar coordinates of the (orthogonal) projection P₀(x, y, 0) of P in the xy -plane.

z is the directed distance from P₀ to P.

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柱面坐標的示意圖 (承上頁)

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Def. (球面坐標系統)

In the spherical coordinate system, a point P(x, y, z)∈ R³ is represented by an ordered triple (ρ, θ, ϕ) with

ρ =|OP| =p

x²+ y²+ z²≥ 0.

θ is the directed angle from the positive x-axis to OP₀, where P₀(x, y, 0) is the (orthogonal) projection of P in the xy -plane.

ϕ is the directed angle from the positive z-axis to OP.

(0≤ ϕ ≤ π)

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球面坐標的示意圖 (承上頁)

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Solutions of Example 5

(a) Since ρ² = x²+ y²+ z² and z = ρ cos ϕ for 0≤ ϕ ≤ π, it follows that

x²+ y²= z²=⇒ (x²+ y²+ z²)− 2z²= 0

=⇒ ρ²− 2ρ²cos²ϕ = 0 =⇒ 1 − 2 cos²ϕ = 0

=⇒ cos ϕ = ± 1

√2 =⇒ ϕ = π

4 or ϕ = 3π 4 . (b) Since ρ² = x²+ y²+ z² and z = ρ cos ϕ, it follows that

x²+ y²+ z²− 4z = 0 =⇒ ρ²− 4ρ cos ϕ = 0

=⇒ ρ(ρ − 4 cos ϕ) = 0

=⇒ ρ = 4 cos ϕ for 0≤ ϕ ≤ π 2.

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Example 5 的示意圖

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Chapter 9 Vectors and the Geometry of Space (向量與空間幾何)

Chapter 9

Vectors and the Geometry of Space (向量與空間幾何)

Hung-Yuan Fan (范洪源)

本章預定授課範圍

9.0 Definitions and Preliminaries 9.6 Surfaces in Space

9.7 Cylindrical and Spherical Coordinates

Section 9.0

Definitions and Preliminaries

(定義和預備知識)

Vectors in the Plane (平面向量)

Definitions and Notations (1/2)

Definitions and Notations (2/2)

Vectors in Space (空間向量)

Vectors in Space

Definitions in R

(1/2)

Definitions in R

(2/2)

向量 −→ PQ 的示意圖 (承上頁)

The Dot Product of Vectors (向量內積)

Def. (向量內積的定義)

Some Special Vectors

Cross Product of Vectors in R

Def. (空間向量的外積)

Notes

外積的示意圖 (承上頁)

Section 9.6 Surfaces in Space

(空間中的曲面)

Type I: Cylindrical Surfaces (柱狀曲面或是柱面)

柱面的示意圖 (承上頁)

Example

圓柱曲面的示意圖 (承上頁)

Example 1 的示意圖

Type II: Quadratic Surfaces (二次曲面)

Type II: Quadratic Surfaces

Ellipsoid 的示意圖 (承上頁)

Example 4 的示意圖

Type II: Quadratic Surfaces

Hyperboloid of One Sheet 的示意圖

Type II: Quadratic Surfaces

Hyperboloid of Two Sheets 的示意圖

Example 2 的示意圖

Type II: Quadratic Surfaces

Elliptic Cone 的示意圖

Type II: Quadratic Surfaces

Elliptic Paraboloid 的示意圖

Example 3 的示意圖

Type II: Quadratic Surfaces

Hyperbolic Paraboloid 的示意圖

Section 9.7

Cylindrical and Spherical Coordinates (柱面坐標與球面坐標)

Def. (柱面坐標系統)

柱面坐標的示意圖 (承上頁)

Def. (球面坐標系統)

球面坐標的示意圖 (承上頁)

Solutions of Example 5

Example 5 的示意圖

Thank you for your attention!