Chapter 9 Vectors and the Geometry of Space Department of - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . . Chapter 9 Vectors and the Geometry of Space Department of Mathematics, National Taiwan Normal University, Taiwan Spring 2019 Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . . . 1/60 ( 向量與空間幾何 ) Hung-Yuan Fan ( 范洪源 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . 9.0 Defjnitions and Preliminaries 9.6 Surfaces in Space 9.7 Cylindrical and Spherical Coordinates Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . . . . 2/60 本章預定授課範圍 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . Section 9.0 Defjnitions and Preliminaries Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . . 3/60 ( 定義和預備知識 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Two-Dimensional Euclidean (Vector) Space Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 4/60 Vectors in the Plane ( 平面向量 ) R 2 = { ( v 1 , v 2 ) | v 1 , v 2 ∈ R } = { v = ⟨ v 1 , v 2 ⟩ | v is a vector ( 向量 ) } Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Defjnitions and Notations (1/2) Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . 5/60 . . . . Vector Addition ( 向量加法 ): ∀ v , u ∈ R 2 . v + u = ⟨ v 1 , v 2 ⟩ + ⟨ u 1 , u 2 ⟩ = ⟨ v 1 + u 1 , v 2 + u 2 ⟩ Scalar Multiplication ( 純量乘法 ): ∀ c ∈ R and v ∈ R 2 . cv = c ⟨ v 1 , v 2 ⟩ = ⟨ cv 1 , cv 2 ⟩ Length or Norm ( 範數 ) of a Vector: � ∀ v ∈ R 2 . ∥ v ∥ = ∥⟨ v 1 , v 2 ⟩∥ = v 2 1 + v 2 2 ≥ 0 v ∈ R 2 is a unit vector ( 單位向量 ) if ∥ v ∥ = 1 . Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Defjnitions and Notations (2/2) Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . 6/60 . . . . If i = ⟨ 1 , 0 ⟩ and j = ⟨ 0 , 1 ⟩ are standard unit vectors in R 2 , then ∀ v ∈ R 2 . v = ⟨ v 1 , v 2 ⟩ = v 1 i + v 2 j If v is represented by the directed line segment from P ( p 1 , p 2 ) to Q ( q 1 , q 2 ) , then it has the component form ( 分量形式 ) v = ⟨ v 1 , v 2 ⟩ = ⟨ q 1 − p 1 , q 2 − p 2 ⟩ ∈ R 2 . Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Three-Dimensional Euclidean (Vector) Space Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 7/60 Vectors in Space ( 空間向量 ) R 3 = { ( v 1 , v 2 , v 3 ) | v 1 , v 2 , v 3 ∈ R } = { v = ⟨ v 1 , v 2 , v 3 ⟩ | v is a vector in space } Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Vectors in Space Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 8/60 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Length or Norm of a Vector: Chapter 9, Calculus B . . . . . . . . . . . . 9/60 . . . . . . . . . . Defjnitions in R 3 (1/2) Vector Addition ( 向量加法 ): v + u = ⟨ v 1 , v 2 , v 3 ⟩ + ⟨ u 1 , u 2 , u 3 ⟩ ∀ v , u ∈ R 3 . = ⟨ v 1 + u 1 , v 2 + u 2 , v 3 + u 3 ⟩ Scalar Multiplication ( 純量乘法 ): ∀ c ∈ R and v ∈ R 3 . cv = c ⟨ v 1 , v 2 , v 3 ⟩ = ⟨ cv 1 , cv 2 , cv 3 ⟩ � ∀ v ∈ R 3 . v 2 1 + v 2 2 + v 2 ∥ v ∥ = ∥⟨ v 1 , v 2 , v 3 ⟩∥ = 3 ≥ 0 v ∈ R 3 is a unit vector if ∥ v ∥ = 1 . Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . If v is represented by the directed line segment from Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . 10/60 . . . . Defjnitions in R 3 (2/2) If i = ⟨ 1 , 0 , 0 ⟩ , j = ⟨ 0 , 1 , 0 ⟩ and k = ⟨ 0 , 0 , 1 ⟩ are standard unit vectors in R 3 , then ∀ v ∈ R 3 . v = ⟨ v 1 , v 2 , v 3 ⟩ = v 1 i + v 2 j + v 3 k P ( p 1 , p 2 , p 3 ) to Q ( q 1 , q 2 , q 3 ) , then it has the component form v = ⟨ v 1 , v 2 , v 3 ⟩ = ⟨ q 1 − p 1 , q 2 − p 2 , q 3 − p 3 ⟩ ∈ R 3 . Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

向量 − → . . . . . . . . . . . . . . . . . Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . . 11/60 PQ 的示意圖 ( 承上頁 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 12/60 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . (3) In some textbooks, the dot product is also called the inner product of vectors. Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . 13/60 . . . . . The Dot Product of Vectors ( 向量內稽 ) Def. ( 向量內稽的定義 ) (1) The dot product of u = ⟨ u 1 , u 2 ⟩ and v = ⟨ v 1 , v 2 ⟩ is u • v = u 1 v 1 + u 2 v 2 ∈ R . (2) The dot product of u = ⟨ u 1 , u 2 , u 3 ⟩ and v = ⟨ v 1 , v 2 , v 3 ⟩ is u • v = u 1 v 1 + u 2 v 2 + u 3 v 3 ∈ R . Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 14/60 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . Some Special Vectors v Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . 15/60 . . . . . Let u and v be vectors in R 2 or R 3 . ∥ v ∥ is the unit vector in the direction of v ̸ = 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 . Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 16/60 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . i j k . Chapter 9, Calculus B . 17/60 . . . . . . . . . . . . . . . . . . . . . . Cross Product of Vectors in R 3 Def. ( 空間向量的外積 ) The cross product of u = ⟨ u 1 , u 2 , u 3 ⟩ and v = ⟨ v 1 , v 2 , v 3 ⟩ is � � � � ( 對第一列作行列式降階 !) � � u × v = � u 1 u 2 u 3 � � � � v 1 v 2 v 3 � � � � � � � � u 2 u 3 � � u 1 u 3 � � u 1 u 2 � = � i − � j + � k . � � � � � � � v 2 v 3 � v 1 v 3 � v 1 v 2 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . Notes orthogonal to u and v , respectively. opposite directions. Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . . 18/60 u × v is a vector in R 3 , but u • v is a scalar. ( u × v ) • u = 0 = ( u × v ) • v , i.e., the vector u × v is v × u = − ( u × v ) , i.e., they are parallel vectors, but in the Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 19/60 外積的示意圖 ( 承上頁 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 20/60 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 21/60 Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . Section 9.6 Surfaces in Space Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . . 22/60 ( 空間中的曲面 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . If the line L is not parallel to the plane containing a curve C , then is a cylindrical surface, or simply a cylinder. The lines parallel to L are rulings. Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . . 23/60 Type I: Cylindrical Surfaces ( 柱狀曲面或是柱面 ) S = { ℓ | ℓ is a line parallel to L and intersecting C } C is the generating curve of S . Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan

. . . . . . . . . . . . . . . . . . Chapter 9, Calculus B . . . . . . . . . . . . . . . . . . . . . . 24/60 柱面的示意圖 ( 承上頁 ) Hung-Yuan Fan ( 范洪源 ), Dep. of Math., NTNU, Taiwan