Engineering Tripos Part IIB, 4M12: Partial Differential Equations & Variational Methods (shared with IIA), 2020-21
Module Leader
Lecturers
Dr J S Biggins and Prof P Davidson
Timing and Structure
Lent term. 16 lectures (including examples classes). Assessment: 100% exam
Aims
The aims of the course are to:
- provide an introduction to the various classes of PDE and the physical nature of their solution
- demonstrate how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the various types of PDE and the physical nature of their solutions.
- understand various solution methods for PDEs and be able to apply these to a range of problems.
- understand the formulation of various physical problems in terms of variational statements
- estimate solutions using trial functions and direct minimisation;
- calculate an Euler-Lagrange differential equation from a variational statement, and to find the corresponding natural boundary conditions;
- perform vector manipulations using suffix notation.
Content
Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis
Suffix notation and the summation convention (2L Dr J S Biggins)
Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes’ theorem and the divergence theorem.
Variational methods in engineering analysis (6L DrJ S Biggins)
Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.
Partial Differential Equations (8L Prof. P. A. Davidson)
What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. solving the diffusion equation. Solving the wave equation. Solving the Laplace equation.
Booklists
Please refer to the Booklist for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 01/09/2020 10:44
Engineering Tripos Part IIB, 4M12: Partial Differential Equations & Variational Methods (shared with IIA), 2018-19
Module Leader
Lecturers
Dr J S Biggins and Prof P Davidson
Timing and Structure
Lent term. 16 lectures (including examples classes). Assessment: 100% exam
Aims
The aims of the course are to:
- provide an introduction to the various classes of PDE and the physical nature of their solution
- demonstrate how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the various types of PDE and the physical nature of their solutions.
- understand various solution methods for PDEs and be able to apply these to a range of problems.
- understand the formulation of various physical problems in terms of variational statements
- estimate solutions using trial functions and direct minimisation;
- calculate an Euler-Lagrange differential equation from a variational statement, and to find the corresponding natural boundary conditions;
- perform vector manipulations using suffix notation.
Content
Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis
Suffix notation and the summation convention (2L Dr J S Biggins)
Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes’ theorem and the divergence theorem.
Variational methods in engineering analysis (6L DrJ S Biggins)
Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.
Partial Differential Equations (8L Prof. P. A. Davidson)
What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. solving the diffusion equation. Solving the wave equation. Solving the Laplace equation.
Booklists
Please see the Booklist for Group M Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 17/08/2018 21:31
Engineering Tripos Part IIB, 4M12: Partial Differential Equations & Variational Methods (shared with IIA), 2019-20
Module Leader
Lecturers
Dr J S Biggins and Prof P Davidson
Timing and Structure
Lent term. 16 lectures (including examples classes). Assessment: 100% exam
Aims
The aims of the course are to:
- provide an introduction to the various classes of PDE and the physical nature of their solution
- demonstrate how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the various types of PDE and the physical nature of their solutions.
- understand various solution methods for PDEs and be able to apply these to a range of problems.
- understand the formulation of various physical problems in terms of variational statements
- estimate solutions using trial functions and direct minimisation;
- calculate an Euler-Lagrange differential equation from a variational statement, and to find the corresponding natural boundary conditions;
- perform vector manipulations using suffix notation.
Content
Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis
Suffix notation and the summation convention (2L Dr J S Biggins)
Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes’ theorem and the divergence theorem.
Variational methods in engineering analysis (6L DrJ S Biggins)
Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.
Partial Differential Equations (8L Prof. P. A. Davidson)
What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. solving the diffusion equation. Solving the wave equation. Solving the Laplace equation.
Booklists
Please see the Booklist for Group M Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 31/05/2019 12:16
Engineering Tripos Part IIB, 4M12: Partial Differential Equations & Variational Methods (shared with IIA), 2025-26
Module Leader
Lecturers
Timing and Structure
Lent term. 16 lectures (including examples classes). Assessment: 100% exam
Aims
The aims of the course are to:
- provide an introduction to the various classes of PDE and the physical nature of their solution
- demonstrate how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the various types of PDE and the physical nature of their solutions.
- understand various solution methods for PDEs and be able to apply these to a range of problems.
- understand the formulation of various physical problems in terms of variational statements
- estimate solutions using trial functions and direct minimisation;
- calculate an Euler-Lagrange differential equation from a variational statement, and to find the corresponding natural boundary conditions;
- perform vector manipulations using suffix notation.
Content
Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis
Suffix notation and the summation convention (2L Prof J S Biggins)
Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes’ theorem and the divergence theorem.
Variational methods in engineering analysis (6L Prof J S Biggins)
Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.
Partial Differential Equations (8L Dr J Li)
What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. Typical solution techniques and example solutions for simple geometries.
Booklists
Please refer to the Booklist for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 04/06/2025 13:33
Engineering Tripos Part IIB, 4M12: Partial Differential Equations & Variational Methods (shared with IIA), 2024-25
Module Leader
Lecturers
Timing and Structure
Lent term. 16 lectures (including examples classes). Assessment: 100% exam
Aims
The aims of the course are to:
- provide an introduction to the various classes of PDE and the physical nature of their solution
- demonstrate how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the various types of PDE and the physical nature of their solutions.
- understand various solution methods for PDEs and be able to apply these to a range of problems.
- understand the formulation of various physical problems in terms of variational statements
- estimate solutions using trial functions and direct minimisation;
- calculate an Euler-Lagrange differential equation from a variational statement, and to find the corresponding natural boundary conditions;
- perform vector manipulations using suffix notation.
Content
Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis
Suffix notation and the summation convention (2L Prof J S Biggins)
Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes’ theorem and the divergence theorem.
Variational methods in engineering analysis (6L Prof J S Biggins)
Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.
Partial Differential Equations (8L Dr J Li)
What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. Typical solution techniques and example solutions for simple geometries.
Booklists
Please refer to the Booklist for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 31/05/2024 10:27
Engineering Tripos Part IIB, 4M12: Partial Differential Equations & Variational Methods (shared with IIA), 2023-24
Module Leader
Lecturers
Timing and Structure
Lent term. 16 lectures (including examples classes). Assessment: 100% exam
Aims
The aims of the course are to:
- provide an introduction to the various classes of PDE and the physical nature of their solution
- demonstrate how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the various types of PDE and the physical nature of their solutions.
- understand various solution methods for PDEs and be able to apply these to a range of problems.
- understand the formulation of various physical problems in terms of variational statements
- estimate solutions using trial functions and direct minimisation;
- calculate an Euler-Lagrange differential equation from a variational statement, and to find the corresponding natural boundary conditions;
- perform vector manipulations using suffix notation.
Content
Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis
Suffix notation and the summation convention (2L Prof J S Biggins)
Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes’ theorem and the divergence theorem.
Variational methods in engineering analysis (6L Prof J S Biggins)
Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.
Partial Differential Equations (8L Dr J Li)
What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. Typical solution techniques and example solutions for simple geometries.
Booklists
Please refer to the Booklist for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 19/09/2023 09:58
Engineering Tripos Part IIB, 4M12: Partial Differential Equations & Variational Methods (shared with IIA), 2022-23
Module Leader
Lecturers
Dr J S Biggins and Prof P Davidson
Timing and Structure
Lent term. 16 lectures (including examples classes). Assessment: 100% exam
Aims
The aims of the course are to:
- provide an introduction to the various classes of PDE and the physical nature of their solution
- demonstrate how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the various types of PDE and the physical nature of their solutions.
- understand various solution methods for PDEs and be able to apply these to a range of problems.
- understand the formulation of various physical problems in terms of variational statements
- estimate solutions using trial functions and direct minimisation;
- calculate an Euler-Lagrange differential equation from a variational statement, and to find the corresponding natural boundary conditions;
- perform vector manipulations using suffix notation.
Content
Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis
Suffix notation and the summation convention (2L Dr J S Biggins)
Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes’ theorem and the divergence theorem.
Variational methods in engineering analysis (6L DrJ S Biggins)
Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.
Partial Differential Equations (8L Prof. P. A. Davidson)
What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. solving the diffusion equation. Solving the wave equation. Solving the Laplace equation.
Booklists
Please refer to the Booklist for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 24/05/2022 13:14
Engineering Tripos Part IIB, 4M12: Partial Differential Equations & Variational Methods (shared with IIA), 2021-22
Module Leader
Lecturers
Dr J S Biggins and Prof P Davidson
Timing and Structure
Lent term. 16 lectures (including examples classes). Assessment: 100% exam
Aims
The aims of the course are to:
- provide an introduction to the various classes of PDE and the physical nature of their solution
- demonstrate how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the various types of PDE and the physical nature of their solutions.
- understand various solution methods for PDEs and be able to apply these to a range of problems.
- understand the formulation of various physical problems in terms of variational statements
- estimate solutions using trial functions and direct minimisation;
- calculate an Euler-Lagrange differential equation from a variational statement, and to find the corresponding natural boundary conditions;
- perform vector manipulations using suffix notation.
Content
Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis
Suffix notation and the summation convention (2L Dr J S Biggins)
Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes’ theorem and the divergence theorem.
Variational methods in engineering analysis (6L DrJ S Biggins)
Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.
Partial Differential Equations (8L Prof. P. A. Davidson)
What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. solving the diffusion equation. Solving the wave equation. Solving the Laplace equation.
Booklists
Please refer to the Booklist for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 20/05/2021 07:52
Engineering Tripos Part IIB, 4M9: Surveying Field Course, 2017-18
Module Leader
Timing and Structure
Long Vacation between Part IIA and Part IIB. 2 - 15 July 2017 for 2017/18. and 1 - 14 July for 2018/19 -Assessment: 100% coursework
Prerequisites
Surveying experience, e.g. from IIA Engineering Area Activity or Fieldwork project.
Aims
The aims of the course are to:
- give students experience in surveying to a high accuracy, on a larger scale (and at greater altitude) than is possible near Cambridge.
Objectives
As specific objectives, by the end of the course students should be able to:
- plan the work for a complex setting-out exercise.
- know how to use high-accuracy and long-range surveying equipment.
- understand the role of GNSS in modern survey.
- know the calculation methods needed for the reduction of three-dimensional survey data.
- have experience in leading a survey team, and the planning of logistics.
- understand the effects of small errors in measurement, and how to minimise their effects.
- understand the need for long-term record keeping, and the information to be recorded.
Content
This module gives students experience in surveying to a high accuracy, on a larger scale than is possible near Cambridge. The exercise includes three-dimensional position-fixing and setting-out in a hilly location, and involves the use of first-order surveying instruments and precise computation.
Throughout the course, short lectures will be given as necessary to explain the theory needed for the practical work in hand. Topics covered include: geoids, ellipsoids, projections and grids; the theory and practice of GNSS, including the verification of Geoid models; reduction of angles and distances; least-squares adjustment.
The course has a capacity of 16. If over-subscribed, a ballot will be held in May, but with preference given to Civil Engineering students.
Coursework
The Course runs continuously over a two week period, and includes the following:
- Exercise planning and siting of control stations;
- Fixing of control stations using GNSS;
- High-accuracy traversing and resectioning;
- Fixing of heights by precise digital levelling and trigonometric heighting;
- Long-range distance measurement;
- Three-dimensional setting out;
- Adjustment, computation and record keeping.
The output of this course will be a set of numerical calculations leading to the setting-out of one or more points in the field. Since incorrect answers will be systematically eliminated from this result, assessment will be based on the course demonstrators' estimation of each student's ability to:
- Take accurate readings efficiently with the equipment provided;
- Make a neat and decipherable record of other students' readings;
- Produce accurate and well laid-out calculations;
- Check the calculations of others;
- Plan and manage the activities of the team;
- Generally contribute to the efficiency and productivity of the team.
Booklists
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
P7
Awareness of quality issues.
P8
Ability to apply engineering techniques taking account of a range of commercial and industrial constraints.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 24/08/2017 15:52
Engineering Tripos Part IIB, 4G6: Cellular & Molecular Biomechanics, 2019-20
Module Leader
Lecturers
Prof V Deshpande and Prof N Fleck
Timing and Structure
Michaelmas term. 14 lectures + 2 examples classes. Assessment: 100% exam
Prerequisites
3C7 useful.
Aims
The aims of the course are to:
- deal with the relation between microstructure of and properties such as strength, stiffness and actuation capability of natural materials such as cells and tissues and their properties, including stiffness.
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the relation between micro-structure of soft biological materials and their mechanical properties.
- have a working understanding of the various components within plant and animal cells with a more detailed knowledge of the cytoskeletal components.
- understand the origins of the mechanical forces generated due to the polymerization of cytoskeletal proteins and derive the key equations.
- develop an understanding of muscles as actuators at the tissue, cell and protein length scales.
Content
Overview Lecture (Prof N. A. Fleck 1L)
The microstructure of the cell – animal cells, plant cells and the sub-cell building materials.
Mechanical Properties of Soft Solids (4L) (Prof. N A Fleck)
- The mechanical properties of natural materials – property maps
- Bending versus stretching micro-structures and entropic networks
- The notion of persistence length
- Models of stiffness and strength
- Mechanics of skin: stress v. strain responses, toughness and skin injection
The cytoskeleton (4L) (Prof.V. Deshpande)
- Review of basic thermodynamics and kinetics
- Introduction to cytoskeletal components and basics mechanics of the filaments
- Re-organization of the cytoskeletal filaments: polymerization, force generation and an introduction to motility
Muscle Mechanics (5L) (Prof.V. Deshpande)
- Twitch and tetanus and the Hill model
- Structure of the muscle: fibers, fibrils and contractile proteins
- Sources of energy in the muscle- Lohmann reaction
- Huxley Sliding filament model
- Models of myosin
Further notes
Further details and online resources:-
http://www-g.eng.cam.ac.uk/lifesciences/courses.html
Booklists
Please see the Booklist for Group G Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 28/05/2019 15:29
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