Differential Geometry

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Syllabus 1 Unit 1 2. Unit 2 3. Unit 3 4. Unit 4 5. Unit 5 6. OLD Question 7. MCQ test 8. Curriculum Planning
Differential Geometry

Course title: Differential Geometry Nature of the course: Theory
Course no.: Math Ed.537 Credit hours: 3
Level: M.Ed. Teaching hours: 48
Semester: Third

1. Course Description

This course is designed to provide wider knowledge and skills on differential geometry for Math Educators. It comprises a range of skills varies from curves in space to intrinsic and extrinsic properties on surface. This course deals with curves and surfaces in 3-space using the tools of calculus and linear algebra. Topics covered in this course includes local and global properties of curves and surfaces. The course is divided in five major units. It starts with curves in space and then introduce some special curves. Then the course introduces surface and its fundamental form. Finally, the course deals with intrinsic and extrinsic properties on surface.

2. General Objectives

The general objectives of this course are as follows:
  • To utilize the concept of a space curve and its types in problem solving
  • To apply basic results of surface to solve related problems
  • To interpret the fundamental forms of surface
  • To explore and prove local properties on surface
  • To calculate and apply fundamental coefficients of surface in problem solving

Unit I: Curves in Space (10)

Learning Outcomes

  • To understand curves in space, and to find its class
  • To define, derive, and compute tangent line and its related theorems
  • To compute order of contact between curve and surface and apply it in problem solving
  • To explain osculating plane, derive its equation and apply it in problem solving and theorem proof
  • To analyze fundamentals of space curve and derive its equations
  • To define curvature and torsion and apply it in problem solving and theorem proof
  • To state and prove fundamental theorem of space curve

Learning Content

  1. Space curve and its class
  2. Tangent to the space curve
  3. Order of contact
  4. Osculating plane
  5. Fundamentals on space curve
  6. Curvature, torsion and screw curvature
  7. Intrinsic equation

Unit II: Special Curves (9)

Learning Outcomes

  • To explain helix and prove its related theorems
  • To define osculating circle and analyze its properties
  • To analyze osculating sphere and prove its properties and related theorems
  • To compare Evolute and involute and compute its curvature and torsion
  • To state Bertrand curves and prove its properties

Learning Content

  1. Cylindrical helix
  2. Osculating circle and osculating sphere
  3. Evolute and involutes
  4. Bertrand curves and its properties

Unit III: Surface (10)

Learning Outcomes

  • To define surface and find its class
  • To analyze regular point, singular point
  • To explain parameter transformation and prove its geometric significance
  • To analyze tangent plane and normal line and use it in problem solving and theorem proof
  • To explain family of surface, and evaluate characteristic line, envelope, characteristic point and edge of regression, and to prove related theorems
  • To compare ruled surface and its kinds and use it in problem solving and theorem proof
  • To explore developable surface associated with space curves and prove related theorems

Learning Content

  1. Surface and its Class
  2. Regular and Singular Point
  3. Transformation and its geometric significance
  4. Tangent plane and normal line
  5. Family of surface
  6. The ruled surface
  7. Developable surface

Unit IV: Fundamental Forms (9)

Learning Outcomes

  • To interpret first and second fundamental forms geometrically and apply them in proving theorems
  • To calculate first and second fundamental coefficients of surface
  • To prove Weingarten equations
  • To explain direction component and direction coefficient
  • To define family of curves and its differential equation
  • To explore orthogonal trajectories and its differential equation
  • To compare double family of curves and its orthogonality

Learning Content

  1. Fundamental forms of surface
  2. Fundamental coefficients of surface
  3. Weingarten equations
  4. Direction coefficients and related results
  5. Family of curves
  6. Orthogonal trajectories
  7. Double family of curves

Unit V: Properties on Surface (10)

Learning Outcomes

  • To define intrinsic and non-intrinsic properties on surface
  • To explain normal curvature, principal curvature, normal section, principal section, principal direction and derive its differential equation
  • To prove Meusnier’s theorem
  • To define developable surface and prove related theorems
  • To understand line of curvatures and prove related theorems
  • To state and prove Rodrigue’s formula, Monge’s theorem, Euler’s theorem, Joachimsthal’s theorem
  • To compute conjugate direction and prove related theorems
  • To define asymptotic lines and prove related theorems
  • To explore fundamental equation and compute christoffel coefficients
  • To state and derive Gauss characteristic equation
  • To derive Mainardi-codazzi equation

Learning Content

  1. Local non-intrinsic property of surface
    1. Local non-intrinsic property of surface
    2. Normal curvature and related theorems
    3. Meusnier’s theorem
  2. Developable surface
  3. Minimal surface
  4. Line of curvature and its properties
    1. Rodrigue’s formula, Monge’s theorem, Euler’s theorem
    2. Joachimsthal’s theorem
  5. Conjugate direction and its properties
  6. Asymptotic lines and related theorems
  7. The fundamental equation of surface theorem
    1. Gauss characteristic equation
    2. Mainardi-codazzi equation

Note: The figures in the parentheses indicate approximate teaching hours allocated for respective units.

4. Instructional Techniques

The instructor will select the method or methods of instruction most suitable for a topic. It is quite acceptable to select more than one method and combine them into a single period of instruction whenever it is needed. The general and specific instructional techniques are described below.

4.1 General Instructional Techniques:

The following general method of instruction will be adopted according to the need and nature of the lesson:
  • Lecture
  • Demonstration
  • Discussion
  • Group work

4.2 Specific Instructional Techniques

  • Multimedia presentation
  • Project work
  • Group Discussion

5. Evaluation

5.1 Internal Evaluation (40%)

Internal evaluation will be conducted by course teacher based on following activities
Activitiespoints
Attendance 5 marks
Participation in learning activities 5 marks
First assessment (assignment) 10 marks
Second assessment (written test) 10 marks
Third assessment (written test) 10 marks
Total 40 marks

5.2 External Examination (60%)

Examination Division, Office of the Dean, Faculty of Education will conduct final examination at the end of the semester. The number of questions and their types along with their marks allocated to each type will be as follows:
typeitemspoints
Objective questions (multiple choice) (10 x 1) 10 marks
Short answer questions 6 with 2 OR questions (6 x 5) 30 marks
Long answer questions 2 with 1 OR question (2 x10) 20 marks
Total 60 marks

6. Recommended and References

6.1 Recommended Books
  1. Gupta, P. P., Mallik, G. S & Pundir, S. K., (2011). Differential geometry. Meerut: Meerut Pragati Prakashan. (Units I -V)
  2. Koirala S. P, & Dhakal B. P. (2068). Differential geometry. Sunlight Publication, Kirtipur, Nepal. (Units I -V)
6.2 Reference Books
  1. Carmo, M. P. (1976) Differential geometry of curves and surfaces. Englewood Cliffs, NJ: Prentice-Hall (Units I & II)
  2. Lal, B., (1969). The three-dimensional differential geometry. Delhi: Atma Ram and Sons. . (Units I-II)
  3. Lipschutz, M. M., (2005). Theory and problems of differential geometry- Schaum’s outline series. Delhi: Tata McGraw-Hill Publishing Company Ltd. 6.
  4. Wilmore, T. J., (2006). An introduction to differential geometry. Delhi: Oxford University Press.. (Units I -V)

Unit I: Curves in Space (10)

Learning Outcomes

  • To understand curves in space, and to find its class
  • To define, derive, and compute tangent line and its related theorems
  • To compute order of contact between curve and surface and apply it in problem solving
  • To explain osculating plane, derive its equation and apply it in problem solving and theorem proof
  • To analyze fundamentals of space curve and derive its equations
  • To define curvature and torsion and apply it in problem solving and theorem proof
  • To state and prove fundamental theorem of space curve

Learning Content

  1. Space curve and its class
  2. Tangent to the space curve
  3. Order of contact
  4. Osculating plane
  5. Fundamentals on space curve
  6. Curvature, torsion and screw curvature
  7. Intrinsic equation

Test your Understaning! Unit 1

Differential Geometry Unit 1 MCQ Test
Unit II: Special Curves (9)
Learning Outcomes
  • To explain helix and prove its related theorems
  • To define osculating circle and analyze its properties
  • To analyze osculating sphere and prove its properties and related theorems
  • To compare Evolute and involute and compute its curvature and torsion
  • To state Bertrand curves and prove its properties
Learning Content
  1. Cylindrical helix
  2. Osculating circle and osculating sphere
  3. Evolute and involutes
  4. Bertrand curves and its properties
Unit III: Surface (10)
Learning Outcomes
  • To define surface and find its class
  • To analyze regular point, singular point
  • To explain parameter transformation and prove its geometric significance
  • To analyze tangent plane and normal line and use it in problem solving and theorem proof
  • To explain family of surface, and evaluate characteristic line, envelope, characteristic point and edge of regression, and to prove related theorems
  • To compare ruled surface and its kinds and use it in problem solving and theorem proof
  • To explore developable surface associated with space curves and prove related theorems
Learning Content
  1. Surface and its Class
  2. Regular and Singular Point
  3. Transformation and its geometric significance
  4. Tangent plane and normal line
  5. Family of surface
  6. The ruled surface
  7. Developable surface
Unit IV: Fundamental Forms (9)
Learning Outcomes
  • To interpret first and second fundamental forms geometrically and apply them in proving theorems
  • o calculate first and second fundamental coefficients of surface
  • To prove Weingarten equations
  • To explain direction component and direction coefficient
  • To define family of curves and its differential equation
  • To explore orthogonal trajectories and its differential equation
  • To compare double family of curves and its orthogonality
Learning Content
  1. Fundamental forms of surface
  2. Fundamental coefficients of surface
  3. Weingarten equations
  4. Direction coefficients and related results
  5. Family of curves
  6. Orthogonal trajectories
  7. Double family of curves
Unit V: Properties on Surface (10)
Learning Outcomes
  • To define intrinsic and non-intrinsic properties on surface
  • To explain normal curvature, principal curvature, normal section, principal section, principal direction and derive its differential equation
  • To prove Meusnier’s theorem
  • To define developable surface and prove related theorems
  • To understand line of curvatures and prove related theorems
  • To state and prove Rodrigue’s formula, Monge’s theorem, Euler’s theorem, Joachimsthal’s theorem
  • To compute conjugate direction and prove related theorems
  • To define asymptotic lines and prove related theorems
  • To explore fundamental equation and compute christoffel coefficients
  • To state and derive Gauss characteristic equation
  • To derive Mainardi-codazzi equation
Learning Content
  1. Local non-intrinsic property of surface
    1. Local non-intrinsic property of surface
    2. Normal curvature and related theorems
    3. Meusnier’s theorem
  2. Developable surface
  3. Minimal surface
  4. Line of curvature and its properties
    1. Rodrigue’s formula, Monge’s theorem, Euler’s theorem,
    2. Joachimsthal’s theorem
  5. Conjugate direction and its properties
  6. Asymptotic lines and related theorems
  7. The fundamental equation of surface theorem
    1. Gauss characteristic equation
    2. Mainardi-codazzi equation

TU 2076

Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Attempt all questions

Group "A"10 X 1 marks =10

  1. In a point \(P\) of a space curve, which of the following is NOT correct?
    1. Tangent has two points contact at \(P\)
    2. Unique tangent line exists at \(P\)
    3. Tangent at \(P\) is limiting form of secant line
    4. Tangent has one point contact at \(P\)
  2. If a curve has constant curvature, then the following is true
    1. Curvature of \(C\) is zero
    2. Torsion of \(C\) is constant
    3. Torsion of \(C\) is zero
    4. Torsion of \(C\) may or may not be constant
  3. Which of the following is true?
    1. Principal normal to a space curve is parallel to principal normal of its locus of center of osculating circle
    2. Principal normal to a space curve is parallel to principal normal of its locus of center of osculating sphere
    3. Tangent to a space curve is parallel to tangent of its locus of center of osculating circle
    4. Tangent to a space curve is parallel to tangent of its locus of center of osculating sphere
  4. Which of the following is NOT correct?
    1. Proper transformation maps regular point to regular point
    2. Proper transformation maps ordinary point to ordinary point
    3. Proper transformation maps singular point to singular point
    4. Proper transformation has non zero Jacobian
  5. Which of the following is the necessary and sufficient condition for a ruled surface \(\vec R = \vec r(s) + v\vec g(s)\) to be a developable surface?
    1. \([\vec{r}', \vec g, \vec{g}'] = 0\)
    2. \([\vec b, \vec g, \vec{g}'] = 0\)
    3. \([\vec n, \vec g, \vec{g}'] = 0\)
    4. \([\vec r, \vec g, \vec{g}'] = 0\)
  6. Second fundamental form measures
    1. Area
    2. Length
    3. Second fundamental coefficient is invariant under parameter transformation
    4. Second fundamental form is invariant under parameter transformation
  7. Which of the following is value of first fundamental coefficient \(F\) in a surface \(\vec r = (u\cos v, u\sin v, u)\)?
    1. \(1\)
    2. \(0\)
    3. \(u^2\)
    4. \(u^2 - 2\)
  8. Which of the following is the condition for a double family of curves \(Pdu^2 + 2Qdudv + Rdv^2 = 0\) to be orthogonal?
    1. \(EN - 2FM + GL = 0\)
    2. \(ER - 2FQ + GP = 0\)
    3. \(PG - 2QF + QE = 0\)
    4. \(LR - 2MQ + NP = 0\)
  9. Which of the following is the sum of principal curvatures?
    1. First curvature
    2. Mean curvature
    3. Second curvature
    4. Gaussian curvature
  10. Which of the following represents Mainardi–Codazzi equation?
    1. \(L_2 - M_1 = mL - (l-\mu)M - \lambda N\)
    2. \(M_2 - M_1 = mL - (l-\mu)M - \lambda N\)
    3. \(N_2 - L_1 = nL - (m-\nu)M - \mu N\)
    4. \(L_2 - L_1 = nL - (m-\nu)M - \mu N\)

Group "B"6 X 5 marks =30
Short answer questions

  1. Define curvature with suitable example. Also prove that necessary and sufficient condition for a curve to be a straight line is that curvature \(\kappa\) at all points is zero.
  2. Define characteristic curves and envelope of a family of surface. Prove that envelope touches each member of the family.
    OR
    Define regular point and singular point on a surface. Prove that proper transformation maps regular point to regular point.
  3. Define polar developable. Prove that generator of polar developable passes through the center of osculating circle and the edge of regression of the polar developable is the locus of center of osculating sphere.
  4. Define second fundamental form of a surface. Also deduce its differential expression.
  5. Define normal curvature. Deduce its differential equation.
    OR
    Define minimal surface. Prove that a surface \(\vec r = (x, y, \log \cos y - \log a \cos x)\) is minimal.
  6. Find the equation of asymptotic line on a surface \(\vec r = (u \cos v, u \sin v, f(u))\). Also prove that necessary and sufficient condition for parametric curves to be asymptotic line is \(L = 0\).

Group "C"2 X 10 marks = 20
Long answer questions

    1. Define fundamental vectors. Also prove that two space curves having same curvature and torsion are identical.
    2. Define circular helix. Also find its equation.
    OR
    1. Define osculating plane. Also deduce its equation.
    2. Define osculating sphere. Derive the equation of locus of center of osculating sphere.
    1. Define line of curvature. If a curve of intersection of two surfaces is line of curvature on both surfaces then prove that two surfaces cut at constant angle.
    2. Define fundamental equation of surface theory. Also prove that \(\dfrac{\partial^2 E}{\partial v^2} + \dfrac{\partial^2 G}{\partial u^2} = \dfrac{1}{2}(N M_{v}-M N_{v}) + \dfrac{1}{2}(L M_{u}-M L_{u})\).

TU 2077

Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Attempt all questions

Group "A"10 X 1 marks =10

  1. How many point contact with the curve do the osculating plane of a space curve have?
    1. one
    2. two
    3. three
    4. four
  2. Which of the following is necessary and sufficient condition for a curve to be a plane curve?
    1. \([\vec r', \vec r'', \vec r''']=0\)
    2. \(\kappa = 0\)
    3. \([\vec r', \vec r'', \vec r]=0\)
    4. \(\tau = 0\)
  3. What do the torsions of two associate Bertrand curves have?
    1. same sign and their product is constant
    2. same sign but their product is not constant
    3. opposite sign and their product is constant
    4. opposite sign but their product is not constant
  4. The proper transformation maps regular point to
    1. inflection point
    2. saddle point
    3. regular point
    4. singular point
  5. What is called the envelope of single parameter family of planes?
    1. developable surface
    2. polar developable surface
    3. skew-developable surface
    4. non polar developable surface
  6. Which of the following is not true about metric?
    1. Metric is positive definite quadratic form in \(du, dv\)
    2. Metric denotes the measurement of arc length on the surface
    3. Metric denotes the relative measure of arc length
    4. Metric is independent of position on the surface
  7. The necessary and sufficient condition for two directions given by \(Pdu^2 + 2Qdudv + Rdv^2 = 0\) to be orthogonal is
    1. \(ER - 2QF + GP = 0\)
    2. \(ER - QF + GP = 0\)
    3. \(LR - 2QM + NP = 0\)
    4. \(LR - QM + NP = 0\)
  8. The parametric curves of a surface cut at right angle if
    1. \(E = 0\)
    2. \(F = 0\)
    3. \(G = 0\)
    4. \(H = 0\)
  9. What is the necessary and sufficient condition for a surface to be developable?
    1. First curvature should be zero
    2. Mean curvature should be zero
    3. Second curvature should be zero
    4. All curvatures should be zero
  10. What is the necessary and sufficient condition for parametric curves to be asymptotic line?
    1. \(L = 0,\; N = 0,\; M \neq 0\)
    2. \(L = 0,\; N = 0,\; M = 0\)
    3. \(L \neq 0,\; N = 0,\; M = 0\)
    4. \(L = 0,\; N \neq 0,\; M = 0\)

Group "B"6 X 5 marks =30
Short answer questions

  1. Define helix. Also state and derive characteristic property of helix.
  2. Define characteristic curve and edge of regression. Prove that each characteristic touches the edge of regression.
    OR
    Prove that necessary and sufficient condition for a surface to be developable is that Gaussian curvature should be zero.
  3. Define developable surface. Prove that edge of regression of the tangential developable of a space curve is the space curve itself.
  4. Define second fundamental form of a surface. Find second fundamental coefficients of a surface given by \(\vec r = (x, y, 2x + 3y^2)\) at origin.
  5. Define principal directions. State and prove Euler’s theorem.
    OR
    Define envelope of family of surface. Find the envelope of the family of planes given by the equation \(F(x, y, z, a) = 3a^2x - 3ay + z - a^3 = 0\).
  6. What are Christoffel coefficients? Prove Mainardi–Codazzi equation given by \(M_2 - N_1 = nL - (m - v)M - \mu N\).

Group "C"2 X 10 marks = 20
Long answer questions

  1. Define involute and evolute. Find curvature and torsion of the involute.
    OR
    State and prove the Existence Theorem.
    1. Define normal curvature. Deduce its differential equation.
    2. Define asymptotic line with analytic expression. Also find equation of asymptotic line of a surface given by \(\vec r = (u\cos v, u\sin v, v)\).

TU 2079

Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Attempt all questions

Group "A"10 X 1 marks =10

  1. Which of the following is the necessary and sufficient condition for a curve to be a plane curve?
    1. the curvature \(\kappa = 0\) at all points of the curve
    2. the torsion \(\tau = 0\) at all points of the curve
    3. the curvature and torsion are in constant ratio
    4. the curvature and torsion both are not zero
  2. Which of the following is true for the torsion of the two associate Bertrand curves?
    1. they have opposite sign and their product is constant
    2. they have opposite sign but their product is not constant
    3. they have same sign and their product is constant
    4. they have same sign but their product is not constant
  3. Which of the following is the locus of point of intersection of all consecutive characteristic curves?
    1. Evolute
    2. Involute
    3. Envelope
    4. Edge of regression
  4. Which of the following is the necessary and sufficient condition for a ruled surface to be skew?
    1. \([\vec r_s, \vec g_s, \vec g] \ne 0\)
    2. \([\vec r_s, \vec g_s, \vec g] = 0\)
    3. \(\vec g \times \vec g_s = 0\)
    4. \(\vec r\) parallel \(\vec g\)
  5. Which of the following is NOT true about first fundamental form?
    1. It is positive definite quadratic in \(du, dv\)
    2. It is invariant
    3. It measures the arc length on the surface
    4. It denotes the relative measure of arc length
  6. Which of the following is the necessary and sufficient condition that parametric curves be orthogonal?
    1. \(E = 0\)
    2. \(F = 0\)
    3. \(G = 0\)
    4. \(H = 0\)
  7. If \(k\) and \(k_n\) are the curvatures of oblique and normal sections through the same tangent line and \(\theta\) is the angle between these sections, then which of the following is true?
    1. \(k_n = k\cos\theta\)
    2. \(k = k_n\cos\theta\)
    3. \(k_n = k\sin\theta\)
    4. \(k = k_n\sin\theta\)
  8. Which of the following is the intrinsic property of surface?
    1. mean curvature
    2. normal curvature
    3. Gaussian curvature
    4. asymptotic lines
  9. Which of the following is the conjugate direction given by double family \(Pdu^2 + Qdudv + Rdv^2 = 0\)?
    1. \(LR - MQ + NP = 0\)
    2. \(LP - MQ + NR = 0\)
    3. \(ER - FQ + GP = 0\)
    4. \(EP - FQ + GR = 0\)
  10. Which of the following is the necessary and sufficient condition that the parametric curves be asymptotic lines?
    1. \(E = M = 0\)
    2. \(E = F = 0\)
    3. \(L = N = 0\)
    4. \(L = M = 0\)

Group "B"6 X 5 marks =30
Short answer questions

  1. Define osculating plane. Find the equation of the osculating plane at any point on the helix \(x = a\cos t,\; y = a\sin t,\; z = ct\).
  2. Define regular point and singular point on a surface. Prove that proper transformation maps regular point to regular point.
  3. Define edge of regression of developable surface. Prove that the tangents to the edge of regression are generators of developable surface.
    OR
    Define tangent plane and normal to a surface. Find the tangent plane and normal line to the surface given by \(x = u + v,\; y = u - v,\; z = uv\).
  4. Define first fundamental form of the surface. Prove that the first fundamental form of the surface is positive definite in \(du, dv\).
  5. Compute the fundamental coefficients for the Monge’s form of the surface \(\vec r = (x, y, f(x, y))\).
  6. Define asymptotic lines. Obtain the condition for asymptotic lines to be orthogonal.
    OR
    Define conjugate direction. Find the differential equation of conjugate direction.

Group "C"2 X 10 marks = 20
Long answer questions

  1. What do you mean by intrinsic equation of a space curve? State and establish the existence theorem. Why is this theorem considered as the fundamental theorem for space curve?
    OR
    1. Define cylindrical helix. Prove that the necessary and sufficient condition for a curve to be a helix is that its curvature and torsion are in constant ratio.
    2. Define involute. Obtain the curvature of involute of a space curve.
  2. What do you mean by principal curvature? Prove that the normal curvature \(K_n\) at a point on a surface in the direction \(\vec r'(u, v)\) is given in terms of principal curvatures as \(K_n = K_1 \cos^2 \phi + K_2 \sin^2 \phi\), where \(\phi\) is the angle that direction of normal section \((du, dv)\) makes with the principal direction \(dv = 0\).

TU 2080 Old Course

Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Attempt all questions

Group "A"10 X 1 marks =10

  1. Which of the following is the equation of rectifying plane?
    1. \((\vec R - \vec r)\cdot \vec t = 0\)
    2. \((\vec R - \vec r)\cdot \vec n = 0\)
    3. \((\vec R - \vec r)\cdot \vec b = 0\)
    4. \((\vec R - \vec r)\cdot \vec N = 0\)
  2. Which of the following is the center of osculating circle?
    1. \(\vec c = \vec r + \rho \vec t\)
    2. \(\vec c = \vec r - \rho \vec n\)
    3. \(\vec c = \vec r + \rho \vec n\)
    4. \(\vec c = \vec r - \rho \vec b\)
  3. Which of the following is the locus of point of intersection of all consecutive characteristic curves?
    1. evolute
    2. involute
    3. envelope
    4. edge of regression
  4. Which of the following is the necessary and sufficient condition for a ruled surface to be developable?
    1. \([\vec r_s,\vec g_s,\vec g]=0\)
    2. \([\vec r_s,\vec g_s,\vec g]\neq 0\)
    3. \(\vec g \times \vec g_s = 0\)
    4. \(\vec r \parallel \vec g\)
  5. Which of the following is NOT true about First Fundamental Form?
    1. It is positive definite quadratic in \(du,dv\)
    2. It is invariant
    3. It measures the arc length on the surface
    4. It denotes the relative measure of arc length
  6. Which of the following is the necessary and sufficient condition that parametric curves are orthogonal?
    1. \(E=0\)
    2. \(F=0\)
    3. \(G=0\)
    4. \(H=0\)
  7. Which of the following is the differential equation for the line of curvature?
    1. \([L\,du^2 + 2M\,du\,dv + N\,dv^2]=0\)
    2. \([N\,d\bar n,d\bar r]=0\)
    3. \([N,d\!N,d\!r]=0\)
    4. \([N,d\!N,d\!b]=0\)
  8. Which of the following is a non‑intrinsic property of surface?
    1. Gaussian curvature
    2. mean curvature
    3. asymptotic lines
    4. normal curvature
  9. Which of the following is the conjugate direction given by double family \(Pdu^2+Qdudv+Rdv^2=0\)?
    1. \(LR - MQ + NP = 0\)
    2. \(ER - FQ + GP = 0\)
    3. \(LP - MQ + NR = 0\)
    4. \(EP - FQ + GR = 0\)
  10. Which of the following is the necessary and sufficient condition that the parametric curves be asymptotic lines?
    1. \(F=M=0\)
    2. \(E=F=0\)
    3. \(L=N=0\)
    4. \(L=M=0\)

Group "B"6 X 5 marks =30
Short answer questions

  1. Find the curvature and torsion for the curve \(x = a\cos t,\; y = a\sin t,\; z = at\). Is this a plane curve? Justify.
  2. Define cylindrical helix. Prove that the necessary and sufficient condition for a curve to be a helix is that its curvature and torsion are in constant ratio.
  3. Define osculating developable. Prove that the generators of osculating developable of a space curve are tangent to the curve.
    OR
    Prove that the envelope of a surface touches each member of the family at all corresponding points of characteristic curves.
  4. Define first fundamental form. Prove that the first fundamental form is invariant under parametric transformation.
  5. Compute the fundamental coefficients for the conoidal surface \(\vec r = (u\cos v,\; u\sin v,\; f(v))\).
  6. Obtain the equation of osculating plane of asymptotic lines. Show that the osculating plane on asymptotic line is the tangent plane to the surface.
    OR
    Define asymptotic lines. Prove that the necessary and sufficient condition that the parametric curves be asymptotic lines is \(L = N = 0,\; M \neq 0\).

Group "C"2 X 10 marks = 20
Long answer questions

    1. Define Bertrand curves. Prove that the distance between corresponding points of Bertrand curves is constant.
    2. Define osculating developable. Prove that the generators of osculating developable of a space curve are tangent to the curve.
    OR
    1. Define involute. Derive the equation of involute.
    2. Prove that the edge of regression of polar developable of a space curve is the locus of center of spherical curvature.
  2. What do you mean by developable surface? Prove that the necessary and sufficient condition for a surface to be developable is that its Gaussian curvature should be zero.

TU 2080 New Course

Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Attempt all questions

Group "A"10 X 1 marks =10

  1. What is the class of the space curve \(x = t^3,\; y = \sin t,\; z = t^{8/3}\)?
    1. 2
    2. 3
    3. 4
    4. 5
  2. Which of the following line or curve has zero curvature?
    1. Straight line
    2. Parabola
    3. Circle
    4. Hyperbola
  3. Which of the following statement is not true for circular helix?
    1. The ratio of curvature and torsion is constant
    2. The curvature at any point of a curve is equal to the torsion at that point
    3. The principal normal is everywhere perpendicular to the axis
    4. The tangent to the curve always makes a constant angle
  4. Which of the following is the necessary and sufficient condition for a ruled surface to be skew?
    1. \([\vec g, \vec g', \vec g'']=0\)
    2. \([\vec g, \vec g', \vec g'']\neq 0\)
    3. \([\vec r_s, \vec g_s, \vec g]=0\)
    4. \([\vec r_s, \vec g_s, \vec g]\neq 0\)
  5. Which of the following is not correct?
    1. Each characteristic touches the edge of regression
    2. The envelope of surface touches each member of the family
    3. The totality of characteristic points for different values of the parameter forms a curve called the characteristic curve of the surface
    4. The tangents to the edge of regression are generators of developable surface
  6. Which of the following is the planar point?
    1. \(LN - M^2 = 0,\; 2 + M^2 + N^2 = 0\)
    2. \(LN - M^2 = 0,\; L^2 + M^2 + N^2 \neq 0\)
    3. \(LN - M^2 = 0\)
    4. \(LN - M^2 \neq 0\)
  7. A surface in space is uniquely determined by certain local invariant quantities called
    1. Curvature and torsion
    2. Mean curvature and Gaussian curvature
    3. Line of curvature
    4. The first and second fundamental form
  8. Let \((l,m,n)\) and \((l',m',n')\) be two directions at any point \(P\) on the surface. When will these two directions conjugate?
    1. \(lFl' + f(m'l' + lm') + Gm'm = 0\)
    2. \(lFm + f(l'm + lm') + Gm'm' = 0\)
    3. \(lFl' + m(m'l' + lm') + nn' = 0\)
    4. \(lmF + lm'(m'/l' + m') + Nl' = 0\)
  9. Which of the following is true for asymptotic direction?
    1. The Gaussian curvature is zero
    2. The normal curvature is zero
    3. The mean curvature is zero
    4. The principal curvature is zero
  10. Which of the following is the expression for specific curvature?
    1. \(\dfrac{EN - 2FM + GL}{EG - F^2}\)
    2. \(\dfrac{LN - M^2}{EG - F^2}\)
    3. \(\dfrac{EN - 2FM + GL}{2(EG - F^2)}\)
    4. \(\dfrac{LN - M^2}{EG - F^2}\)

Group "B"6 X 5 marks =30
Short answer questions

  1. Define osculating plane and derive its equation.
  2. Define torsion. Prove that the necessary and sufficient condition for a given curve to be a plane curve is that \([\vec r\,', \vec r\,'', \vec r\,''']=0\) at all points.
  3. Define envelope of a family of surface. Prove that the envelope of surface touches each member of the family at all points of characteristic curves.
    OR
    Show that the generator of polar developable passes through the center of osculating circle and the edge of regression of polar developable of a space curve is the locus of center of spherical curvature.
  4. Define fundamental magnitudes of a surface. Find the fundamental magnitudes for the conoidal surface \(\vec r = (u\cos v, u\sin v, f(v))\).
  5. Prove that a necessary and sufficient condition for a surface to be developable is that its Gaussian curvature should be zero.
    OR
    Deduce the Mainardi–Codazzi equations
    \(L_2 - M_1 = nL - (l-\mu)M - \lambda N\) and
    \(M_2 - N_1 = nL - (m-\nu)M - \mu N\).
  6. Define edge of regression. Find the edge of regression of the family of planes given by the equation \(F(x,y,z,a): 3a^2x - 3ay + z - a^3 = 0\).

Group "C"2 X 10 marks = 20
Long answer questions

  1. Define osculating circle. State and prove the properties of the locus of center of osculating circle.
    OR
    State and prove the four properties of Bertrand curves.
    1. State and prove Joachimsthal’s theorem.
    2. Define second fundamental form of a surface and give its geometrical interpretation.

TU 2081 Old Course

Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Attempt all questions

Group "A"10 X 1 marks =10

  1. Which of the following is the equation of rectifying plane?
    1. \((\vec R - \vec r)\cdot \vec b = 0\)
    2. \((\vec R - \vec r)\cdot \vec n = 0\)
    3. \((\vec R - \vec r)\cdot \vec t = 0\)
    4. \((\vec R - \vec r)\cdot \vec N = 0\)
  2. Which of the following surface has an essential singularity?
    1. Sphere
    2. Cylinder
    3. Cone
    4. Ellipsoid
  3. Which of the following statement is not true?
    1. The principal normal is everywhere perpendicular to the axis
    2. The ratio of curvature to the torsion of curve at any point is constant
    3. The tangent to the curve always makes a constant angle
    4. The curvature at any point of a curve is equal as the torsion at that point
  4. Which of the following is not correct?
    1. Each characteristic touches the edge of regression
    2. The envelope of surface touches each member of the family
    3. The totality of characteristic points for different values of the parameter forms a curve called the characteristic curve of the surface
    4. The tangents to the edge of regression are generators of developable surface
  5. Which of the following is the necessary and sufficient condition that parametric curves be orthogonal?
    1. \(E = 0\)
    2. \(F = 0\)
    3. \(G = 0\)
    4. \(H = 0\)
  6. Which of the following is the planar point?
    1. \(LN - M^2 = 0,\; L^2 + M^2 + N^2 = 0\)
    2. \(LN - M^2 = 0,\; L^2 + M^2 + N^2 \neq 0\)
    3. \(LN - M^2 \neq 0\)
    4. \(LN - M^2 = 0\)
  7. Which of the following is not true about metric?
    1. It is positive definite quadratic form in \(du, dv\)
    2. It denotes the relative measure of arc length
    3. It is invariant under parametric transformation
    4. It denotes the measurement of the arc length on the surface
  8. Which of the following is an intrinsic property of surface?
    1. Gaussian curvature
    2. Mean curvature
    3. Normal curvature
    4. Line of curvature
  9. What is the condition for two directions given by \(Pdu^2 + Qdudv + Rdv^2 = 0\) to be conjugate?
    1. \(LR - FQ + GP = 0\)
    2. \(EP - FQ + GR = 0\)
    3. \(LR - MQ + NP = 0\)
    4. \(LP - MQ + NR = 0\)
  10. Which is the necessary and sufficient condition that parametric curves be asymptotic lines?
    1. \(L = M = 0,\; N \neq 0\)
    2. \(M = N = 0,\; L \neq 0\)
    3. \(L = N = 0,\; M \neq 0\)
    4. \(L = M = N = 0\)

Group "B"6 X 5 marks =30
Short answer questions

  1. Define curvature. Prove that the necessary and sufficient condition for a curve to be a straight line is that the curvature is zero at all points of the curve.
  2. Define envelope of family of surface. Prove that the envelope of surface touches each member of the family at all points of characteristic curves.
    OR
    Define edge of regression. Find the edge of regression of the family of planes given by the equation \(F(x,y,z,a): 3a^2x - 3ay + z - a^3 = 0\).
  3. Define first fundamental form of the surface and give its geometrical interpretation.
  4. Prove that a necessary and sufficient condition for a surface to be developable is that Gaussian curvature should be zero.
    OR
    Define principal curvature. Deduce the differential equation of principal curvature.
  5. Show that the generator of polar developable passes through the center of osculating circle and the edge of regression of polar developable of a space curve is the locus of center of spherical curvature.
  6. Define osculating circle. Find the radius and center of the osculating circle.

Group "C"2 X 10 marks = 20
Long answer questions

  1. Define evolute. Deduce the curvature and torsion of the evolute.
    OR
    State and prove the four properties of Bertrand curves.
    1. Define line of curvature. Prove that the necessary and sufficient condition that a curve on a surface be a line of curvature is that the surface normal along the curve forms a developable.
    2. Define asymptotic direction. Show that the asymptotic directions are orthogonal if and only if the surface is minimal.

TU 2081 New Course

Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Attempt all questions

Group "A"10 X 1 marks =10

  1. What is the necessary and sufficient condition for a curve to be a plane curve?
    1. The curvature is zero at all points of the curve
    2. The torsion is zero at all points of the curve
    3. The curvature and torsion are in constant ratio
    4. The curvature varies inversely to the torsion
  2. Which condition is the singularity of a point on a surface?
    1. \(\vec r_u \times \vec r_v \neq 0\)
    2. \(\vec r_u \times \vec r_v = 0\)
    3. \(\vec r_u \cdot \vec r_v \neq 0\)
    4. \(\vec r_{uu} \times \vec r_{vv} = 0\)
  3. Which of the following is not true about Bertrand curves?
    1. The distance between corresponding points of Bertrand curves is constant
    2. Tangent at corresponding points of two Bertrand curves is inclined at constant angle
    3. Curvature and torsion of Bertrand curves are connected by linear relation
    4. The curvature of two associate Bertrand curves has same sign and their product is constant
  4. Which of the following is the necessary and sufficient condition that parametric curves be orthogonal?
    1. \(E = 0\)
    2. \(G = 0\)
    3. \(F = 0\)
    4. \(H = 0\)
  5. Which of the following statement is true?
    1. Each characteristic is normal to the edge of regression
    2. Each characteristic is tangent to the edge of regression
    3. The edge of regression denotes a point of intersection of two consecutive characteristics
    4. The edge of regression is the envelope of one‑parameter family of surface
  6. Which direction coefficient of parametric curve \(v = \text{constant}\)?
    1. \(\left(\frac{1}{G}, 0\right)\)
    2. \(\left(0,\frac{1}{\sqrt{E}}\right)\)
    3. \(\left(0,\frac{1}{\sqrt{G}}\right)\)
    4. \(\left(\frac{1}{\sqrt{E}},0\right)\)
  7. Which of the following is not true about metric?
    1. It is positive definite quadratic form in \(du,dv\)
    2. It denotes the measurement of the arc length on the surface
    3. It is invariant under parametric transformation
    4. It denotes the relative measure of arc length
  8. Which of the following represents the differential equation of asymptotic lines?
    1. \(Edu^2 + 2Fdudv + Gdv^2 = 0\)
    2. \(Edu^2 + Fdudv + Gdv^2 = 0\)
    3. \(Ldu^2 + 2Mdudv + Ndv^2 = 0\)
    4. \(Ldu^2 + Mdudv + Ndv^2 = 0\)
  9. Which is the necessary and sufficient condition that parametric curves be lines of curvature?
    1. \(F = M = 0\)
    2. \(E = L = 0\)
    3. \(G = N = 0\)
    4. \(L = M = N = 0\)
  10. For \(\vec r = (u, v, uv)\), what is the value of \(E\)?
    1. \(u^2 + 1\)
    2. \(uv\)
    3. \(1 + v^2\)
    4. \(1 + u^2 + v^2\)

Group "B"6 X 5 marks =30
Short answer questions

  1. Define involute. Deduce the equation of an involute in space curve.
  2. Define cylindrical helix. Show that the necessary and sufficient condition for a curve to be a helix is that \([\vec r\,', \vec r\,'', \vec r\,'''] = 0\).
  3. Define edge of regression of a family of surface. Show that each characteristic touches the edge of regression.
    OR
    Define tangent plane and normal line. Find the equation of tangent plane and normal line to the surface \(x^2 + y^2 - z = 0\) at a point \(P(1,1,2)\).
  4. Show that the generators of osculating developable of a space curve are tangent to the curve and the edge of regression of osculating developable of a space curve is the curve itself.
  5. Define principal curvature. Deduce the equation of principal curvature.
    OR
    Define asymptotic lines and show that the necessary and sufficient condition that the parametric curves be asymptotic lines is \(L = N = 0,\; M \neq 0\).
  6. Define fundamental magnitudes of the surface. Find the fundamental magnitudes for the Monge’s form of the surface \(\vec r = (x, y, f(x,y))\).

Group "C"2 X 10 marks = 20
Long answer questions

  1. State and prove the Existence theorem of differential geometry.
    OR
    Define curvature and torsion. Prove that a necessary and sufficient condition for a curve to be a plane curve is that the torsion is zero at all points on the plane curve. Find curvature and torsion for a space curve \(x = a(3t - t^3),\; y = 3at^2,\; z = a(3t + t^3)\).
    1. State and prove Rodrigues’ formula.
    2. With the help of Weingarten equations, deduce \(H[\vec{N},\vec N_1,\vec{r_1] = EM - FL\).

Curriculum Implementation Plan

CREDIT HOURS [3]
Upon successful completion of this course, you will earn 3 semester hour of college credit, achieved through [16T+32S] 48 hours of student work. Student work may include, but is not limited to, direct or indirect faculty instruction, listening to class lectures or webinars (synchronous or asynchronous), attending a study group that is assigned by the institution, submitting an academic assignment (quiz, discussion, projects), taking an exam, laboratory work, externship or internship. It also includes preparation time such as for homework, reading and study time.
ASSIGNMENT POLICY
Students are expected to meet all deadlines relative to Quiz, discussions and assignments. The late submission will receive a 10% deduction on full marks.
CIP Summary || Math Ed 537
Date (2026)XLessonActivityAssignment
11-21-231Space curveQDAAssignment
21-28-301Osculating planeQDADiscussion
32-4-61Curvature/TorsionQDAQuiz
42-11-132HelixQDAAssignment
52-18-202Osculating Circle/SphereQDADiscussion
62-25-272Involute/Evolute & Bertrand CurvesQDAQuiz
73-4-63Surface/Class; Regular/Singular; Tangent/NormalQDAAssignment
83-11-133Family of surfaceQDADiscussion
93-18-203Ruled and DevelopableQDAQuiz
103-25-274Fundamental forms of surface coefficientsQDAAssignment
114-1-34Weingarten equations; Direction coefficientsQDADiscussion
124-8-104Orthogonal trajectories; Double family of curvesQDAQuiz
Week 134-15-175PropertiesQDAAssignment
144-22-245Developable and MinimalQDADiscussion
154-29-5/15Theorems: Rodrigue, Euler, Gauss, MainardiQDAQuiz
165-6-8ReviewFinal ReviewQDAReview
QDA: Quiz, Discussion, Q/A demonstration

Lesson 1

LESSON 1 LEARNING OBJECTIVES (LO’s)
After completing this Lesson, students should be able to understand the following concepts:
 Define curve in space
 Give explicit and implicit example of space curve
 Execute and identify the class of space curve
LESSON 1 READING ASSIGNMENT
TEXTBOOK
 Chapter 1: Curves in Space
INSTRUCTOR WEEKLY VIDEO
Review the instructor’s video for this Lesson.
LESSON 1 LESSON ACTIVITY
1. Write five example of space curve in parametric form
2. Write five example of space curve in implicit form
LESSON 1 Assignment
1. Write five example of space curve whose class are respectively 1,2,3,4,5.

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