1 Faizah Al-Harbi

Differential Geometry

MAT 4390

Level 8

( Elective Course )

2 Faizah Al-Harbi

( 4390ريض – الهندسة التفاضلية ) وصف مقرر

Course Description (Differential Geometry – Math 4390 )

محتويات المقرر:

المتجهات المماسية والعمودية. هندسة – يشمل هذا المقرر هندسة المنحنيات فى المستوى: طول القوس

اء إعادة إنش –يه وسري يهعالقات فرين –اإللتواء – اإلنحناء –المنحنيات فى الفراغ: طول القوس

نحناء إلا – المساحات وانحناءات جاوس – األسطح فى الفراغ – المنحنيات باستخدام االنحناء والتقوس

االنحناء – نظرية اجريجوم – الرئيسى والمتوسط والمساحات األدنى. األسطح الهندسية األصلية

. التغير األول لطول القوس – الجيوديسى للمنحنيات على األسطح

يقدم هذا المقرر باللغة اإلنجليزية ملحوظة:

Course Description:

Geometry of Curves in the Plane – Arc Length – Tangential and Normal Vectors – (signed)

Curvature – Reconstruction of a Curve with given Curvature and Arc Length – Evolutes

and Involutes – the Isoperimetric Inequality and Hopf’s Theorem on the Tangential Degree

of an Embedded Closed Curve – Geometry of Curves in the Space – Arc length – Curvature

– Torsion – The Frenet– Serret Equations – Reconstruction of a curve with given curvature

and torsion – Generalized helices – Evolutes and involutes. Surfaces in Space: The first

and second fundamental forms – Area and the Gauss and Codazzi Equations – Gaussian

curvature – developable surfaces – principal curvature – Meunier’s Theorem – surfaces

of constant Gaussian curvature – mean curvature – minimal surfaces – Intrinsic Geometry

of Surfaces – Geodesic curvature of curves on surfaces – First variation of arc length –

The Gauss– Bonnet Theorem and applications.

3 Faizah Al-Harbi

Suggested Textbooks( الكتب المقترحة)

-Dirk Jan Struik, "Lectures on classical differential geometry", Dover Publications.

- Elementary Differential Geometry, Second Edition, Barrett O’Neill, 2006

- Schaum’s outlines. "Differential Geometry", Martin M. Lipschutz, Ph. D., 1969,

McGraw-Hill.

(مراجع إضافية مفيدة)

1. R.S. Millman and G.D. Parker, “Elements of Differential Geometry”, Prentice Hall, 1977.

2.M.P. do Carmo, “Differential geometry of curves and surfaces”, Prentice Hall, 1976

Important Remark for students

You must rely on the proposed references mentioned earlier to examine this course and

not rely solely on was stated in the note , It is only for explanation and clarification .

Best regards

Faizah Al-Harbi

مالحظة هامة للطالبات :

يجب االعتماد على المراجع المقترحة المذكورة سابقا لدراسة هدا المقرر و عدم االعتماد فقط

هذه المذكرة ألنها فقط للشرح و التوضيح . و هللا الموفقعلى ما ذكر في

: فائزه الحربي أستاذة المقرر

4 Faizah Al-Harbi

Lecture 1

Curves in the plane and the space Review :

Symbols : -1

ℝ → is called Set of real numbers , represented by line numbers (we write 𝒙 ∈ ℝ)

ℝ2→ is called the plane ( it is the set of (𝑥, 𝑦) of real numbers ℝ ) .

ℝ3→ is called the three-dimensional vector space ( Elements of ℝ

3are (𝑥, 𝑦, 𝑧) of real

numbers ℝ )

Orthogonal Curvilinear Coordinates

Spherical 𝑟,𝜃,𝜑

Cylindrical 𝜌,𝜑,𝑧

Cartesian 𝑥,𝑦,𝑧

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:ℝ𝟑Vectors -2

Every vector v⃑ in ℝ3has a unique representation of the form

v⃑ = 𝑣1𝑖 + 𝑣2𝑗 + 𝑣3𝑘 ≡ (𝑣1, 𝑣2, 𝑣3)

where 𝑣1, 𝑣2, 𝑣3 ∈ ℝ and { 𝑖 , 𝑗 , 𝑘}is the standard basis of ℝ3

: Remark

1 − 𝑣𝑒𝑐𝑡𝑜𝑟 {𝑠𝑎𝑐𝑙𝑎𝑟 (𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦)𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛

2- The zero vector 0⃑ = (0,0,0)

3-

4-

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Let v⃑ = (𝑣1, 𝑣2, 𝑣3) 𝑎𝑛𝑑 w⃑⃑⃑ = (𝑤1, 𝑤2, 𝑤3) 𝑎𝑛𝑑 u⃑ = (𝑢1, 𝑢2, 𝑢3)

: v⃑ · w⃑⃑⃑ = 𝑣1𝑤1 + 𝑣2𝑤2 + 𝑣3𝑤3 product )or dot ( innerThe scalar*

geometrically, we have v⃑ · w⃑⃑⃑ = |v⃑ | |w⃑⃑⃑ | cos θ ,

orthogonal vectors :*

Remark :

v⃑ . w⃑⃑⃑ = 0⃑ ⟹ {v⃑ = 0⃑

w⃑⃑⃑ = 0⃑

v⃑ ⊥ w⃑⃑⃑

where |v⃑ | = √𝑣12 + 𝑣2

2 + 𝑣32 ⟹ v ⃑⃑ . v⃑ = |v⃑ |2

is the magnitude of the vector v⃑ (and similarly for |w⃑⃑⃑ |) and θ is the angle between

the vectors v⃑ and w⃑⃑⃑ .

v⃑ . w⃑⃑⃑ = w⃑⃑⃑ . v⃑ :is commutative The dot product *

w cosθ =w⃑⃑⃑ .v⃑⃑

|v|is v⃑ on w⃑⃑⃑ of Project*

v⃑ × w⃑⃑⃑ ≡ v⃑ ∧ w⃑⃑⃑ = |

𝑖 𝑗 𝑘

𝑣1 𝑣2 𝑣3

𝑤1 𝑤2 𝑤3

| The vector product:*

geometrically, we have v⃑ × w⃑⃑⃑ = |v⃑ ||w⃑⃑⃑ |sin θ �̂� , where �̂� is unit vector.

v⃑ × w⃑⃑⃑ ≠ w⃑⃑⃑ × v⃑ 𝑎𝑛𝑑 v⃑ × w⃑⃑⃑ = − w⃑⃑⃑ × v⃑

v⃑ × w⃑⃑⃑ = 0⃑ ⟹ {v⃑ = 0⃑

w⃑⃑⃑ = 0⃑

v⃑ ∥ w⃑⃑⃑

w cosθ

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[u⃑ , v⃑ , w⃑⃑⃑ ] ≡ u⃑ . ( v⃑ × w⃑⃑⃑ ) = |

𝑢1 𝑢2 𝑢3

𝑣1 𝑣2 𝑣3

𝑤1 𝑤2 𝑤3

| : product calars ripletThe *

Remark :

1- u⃑ . ( v⃑ × w⃑⃑⃑ ) = w⃑⃑⃑ . ( u⃑ × v⃑ ) = v⃑ . ( w⃑⃑⃑ × u⃑ )

2- u⃑ . ( v⃑ × w⃑⃑⃑ ) = 0⃑ ⟹ {u⃑ = 0⃑ 𝑜𝑟 v⃑ = 0⃑ 𝑜𝑟 w⃑⃑⃑ = 0⃑

v⃑ ∥ w⃑⃑⃑

𝑖𝑓 v⃑ × w⃑⃑⃑ = F⃑ 𝑎𝑛𝑑 u⃑ . F⃑ = 0 ⇒ u⃑ ⊥ F⃑ 𝑡ℎ𝑒𝑛 u⃑ , v⃑ , w⃑⃑⃑ 𝑎𝑟𝑒 𝑖𝑛 𝑜𝑛𝑒 𝑝𝑙𝑎𝑛𝑒

Theorem: [u⃑ , v⃑ , w⃑⃑⃑ ] = 0 if and only if u⃑ , v⃑ , w⃑⃑⃑ are linearly dependent

�̂� =𝐴

|𝐴 | The unit vector :*

vector start from original point: Any The position vector*

�⃑� = 𝑥𝑒1 + 𝑦𝑒2 + 𝑧𝑒3

α = 𝑐𝑜𝑠−1(𝑥

|�⃑� |) , β = 𝑐𝑜𝑠−1(

𝑦

|�⃑� |) , γ = 𝑐𝑜𝑠−1(

𝑧

|�⃑� |)

Remark :

The vector between two point 𝑝1(𝑥1, 𝑦1, 𝑧1), 𝑝2(𝑥2, 𝑦2, 𝑧2) 𝑖𝑛 ℝ3, then

𝑝1𝑝2⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑ = (𝑥2 − 𝑥1)𝑒1 + (𝑦2 − 𝑦1)𝑒2 + (𝑧2 − 𝑧1)𝑒3

|𝑝1𝑝2⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑| = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)

2 + (𝑧2 − 𝑧1)2

he standard basis :T

one in the basis has only vector in which each orthonormal basis A standard basis is a

. in the real vector ) This is called unit vector( nonzero entry, and that entry is equal to 1

consists of the unit vectors , the standard basis ℝ3 space

𝑒1 ≡ 𝑖 = (1,0,0) , 𝑒2 ≡ 𝑗 = (0,1,0) , 𝑒3 ≡ 𝑘 = (0,0,1)

8 Faizah Al-Harbi

Then

|𝑖| = |𝑗| = |𝑘| = 1 and 𝑖 × 𝑖 = 𝑗 × 𝑗 = 𝑘 × 𝑘 = 0⃑

𝑖 × 𝑗 = 𝑘 , 𝑗 × 𝑘 = 𝑖 , 𝑘 × 𝑖 = 𝑗 ,

𝑗 × 𝑖 = −𝑘 , 𝑘 × 𝑗 = −𝑖 , 𝑖 × 𝑘 = −𝑗

Remark :

1- In ℝ3 −space , any three vectors form a basis of space iff

* Linear independent.

* Any vector in ℝ3 written as a linear combination of their .

2- Any three vector represents 3 columns of matrix 𝐴 = [𝑒1 𝑒2 𝑒3] = [1 0 00 1 00 0 1

]

3- When det (𝐴) ≠ 0 then (𝑒1 , 𝑒2 , 𝑒3) are linear independent.

Q: Is the basis of ℝ3 unique?

Example:

1-

9 Faizah Al-Harbi

2- If we have three vectors (v1 , v2 , v3) given by

𝑣1⃑⃑⃑⃑ = 2𝑒1 − 𝑒2 + 𝑒3

𝑣2⃑⃑⃑⃑ = 𝑒2 + 𝑒3

𝑣3⃑⃑⃑⃑ = 𝑒1 + 2𝑒2 + 𝑒3

Prove that : 1- (𝑣1 , 𝑣2 , 𝑣3) are basis of ℝ3

3- write (𝑒1 , 𝑒2 , 𝑒3) as a linear combination of (𝑣1 , 𝑣2 , 𝑣3)

10 Faizah Al-Harbi

11 Faizah Al-Harbi

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Exercises

1- Let u⃑ , v⃑ , w⃑⃑⃑ ∈ ℝ3. By writing both sides in terms of coordinates, show that

u ⃑⃑⃑ × (v⃑ × w⃑⃑⃑ ) = (u⃑ · w⃑⃑⃑ )v⃑ − (u⃑ · v⃑ )w⃑⃑⃑

2- Let 𝐴 = 3𝑒1 − 7𝑒2 + 10𝑒3

�⃑� = −𝑒1 + 3𝑒3

𝐶 = 2𝑒1 + 5𝑒2 − 𝑒3

Find : 1- 𝑐𝑜𝑠(𝐴, 𝐵)

2- 𝐴 × �⃑�

3- 𝐶 . ( 𝐴 × �⃑� )

4- show that (𝐴 , �⃑� , 𝐶 ) form another basis of ℝ3or not ?

5- The angles (𝛼, 𝛽, 𝛾) with (𝑥, 𝑦, 𝑧) directions for �⃑� ?