Continuum Mechanics I

About Course

Continuum Mechanics can be thought of as the grand unifying theory of engineering science. Many of the courses taught in an engineering curriculum are closely related and can be obtained as special cases of the general framework of continuum mechanics. The balance laws of mass, momentum, and energy that are derived in the context of specific material constitutions are natural laws that are natural laws and independent of these contexts. This fact is easily lost on most undergraduate and even some graduate students.

The language of Continuum Mechanics is called Tensor Analysis, you need Software to practice and engage more challenging problems; then Simulation will help you deploy the knowledge gained to design virtually in order to save prototyping costs. In this set of courses, we take you through all these stages to enrich your knowledge. The approach here is to optimize your time so to learn things the shortest way and remain focused on doing engineering with your knowledge.

Engineering is the application of Science to create technology products and services. It is rooted in theory. If you do not organize the learning of theory very well, you end up with full heads and no products as we have been doing. If you leave theory and simply do “practicals”, you end up with half-baked crafts trade – again, no serious products. We are offering you an approach to avoid both extremes and learn, in order to do engineering correctly.

What you will learn here will alter your view about some of the other courses you will take on your way to a degree in engineering. If you do your part, you will be given skills, tools and knowledge that are directed at making you productive people that can change the narrative of dependency and hopelessness that has been Africa’s story.

Course Content

Vectors: Elementary Principles & Computations Practicum
“God made the integers, all else is the work of man.” – Leopold Kronecker 1. A set of linearly independent vectors is a set where one member cannot be expressed as a linear combination of the others. 2. When you have the maximum number of such vectors in a set, all other vectors in that space can be expressed as linear combinations of the members of this set. The set of orthonormal vectors, {?, ?, ?}, we are used to in the Cartesian form is only one kind of such a set. Occasions will arise that will make other linearly independent vectors useful to know. 3. When a set is complete – having the maximum number of linearly independent vectors, it is said to form a basis of the vector space that it spans. These words are codes to express the fact that they can be used to represent any other vector in the space. All that will be needed is the set of scalar weights (or scaling factors) of the basis vectors will represent each vector. 4. These scalars are called components of the specific vectors represented. Once they are found, with the basis in mind, we use them instead of the vectors they represent because analyses are easier done with the components. 5. #4 above can lead to confusing the vector with its matrix representation. The components of a vector are meaningless unless we specify the basis vectors underlying the representation. This is where the vector, and as we shall see later, the tensor objects, significantly differ from the matrices they look like. 6. The number of vectors constituting a basis spanning the space is the dimension of that space. 7. We gain valuable compactness using the index notation and the Summation Convention. Mastering it early is a great advantage for later work. 8. Other topics treated include Coordinate transformations, Dyads and Rotations. General Curvilinear coordinates are introduced as an advanced topic that can be omitted at first reading. 9. The chapter ends with a brief introduction to Software (Mathematica) the we use to avoid tedium and helps to tackle more challenging problems than could be easily done manually or with a calculator. 10. Mathematica is one of two important software for the series of lectures and courses. Students that start early with the software will gain a lot of ground, will find that the subject helps to learn the software and the software makes learning the subject easier. There will be a lot of examples you CANNOT easily do manually. Those that postpone learning the software are already failing! The best time to learn is at the beginning! You gain a lot and should never be behind!

Week One: Vectors & Linear Independence
Week Two: The Summation Convention
Week Three: Coordinate Transformations of Euclidean Spaces
Programming Clinic 1: Programming in the Wolfram Language
Week Four: Basis Vectors Euclidean Point
Week Five: The Vector Space
SSG 321: Self-Test Guide (Chapter 1)
Self Test Instructions

Vectors: Elementary Principles & Computation Practicum – PG 2021 Videos

Vectors: Elementary Principles & Computations Practicum UG 2025 – Videos
“God made the integers, all else is the work of man.” – Leopold Kronecker 1. A set of linearly independent vectors is a set where one member cannot be expressed as a linear combination of the others. 2. When you have the maximum number of such vectors in a set, all other vectors in that space can be expressed as linear combinations of the members of this set. The set of orthonormal vectors, {?, ?, ?}, we are used to in the Cartesian form is only one kind of such a set. Occasions will arise that will make other linearly independent vectors useful to know. 3. When a set is complete – having the maximum number of linearly independent vectors, it is said to form a basis of the vector space that it spans. These words are codes to express the fact that they can be used to represent any other vector in the space. All that will be needed is the set of scalar weights (or scaling factors) of the basis vectors will represent each vector. 4. These scalars are called components of the specific vectors represented. Once they are found, with the basis in mind, we use them instead of the vectors they represent because analyses are easier done with the components. 5. #4 above can lead to confusing the vector with its matrix representation. The components of a vector are meaningless unless we specify the basis vectors underlying the representation. This is where the vector, and as we shall see later, the tensor objects, significantly differ from the matrices they look like. 6. The number of vectors constituting a basis spanning the space is the dimension of that space. 7. We gain valuable compactness using the index notation and the Summation Convention. Mastering it early is a great advantage for later work. 8. Other topics treated include Coordinate transformations, Dyads and Rotations. General Curvilinear coordinates are introduced as an advanced topic that can be omitted at first reading. 9. The chapter ends with a brief introduction to Software (Mathematica) the we use to avoid tedium and helps to tackle more challenging problems than could be easily done manually or with a calculator. 10. Mathematica is one of two important software for the series of lectures and courses. Students that start early with the software will gain a lot of ground, will find that the subject helps to learn the software and the software makes learning the subject easier. There will be a lot of examples you CANNOT easily do manually. Those that postpone learning the software are already failing! The best time to learn is at the beginning! You gain a lot and should never be behind!

Tensor Algebra: Properties of a Tensor
“Continuum Mechanics may appear as a fortress surrounded by the walls of tensor notation” E. Tadmore, et. al. 1. The word “Tensor” applies to virtually all the quantities encountered in Engineering. Scalars and Vectors are zeroth and first order tensors. However, the word, with no prefix, refers to a second-order tensor. 2. For an object to be proved to be a tensor we need to show that it transforms a vector and its output is also a vector. Secondly, that transformation must be linear. 3. A tensor can be expressed in Component Form. The vector itself is more than its components as components presume reference to particular coordinate system. Outside that the numbers mean nothing. 4. For any tensor, certain scalar-valued functions are characteristic of the tensor, independent of coordinate systems. These are usually the targets of computations of any tensor. They are Principal Invariants. 5. Tensors can be decomposed additively or multiplicatively to simpler tensors. The goal is to make analysis easier and gain valuable insight by removing parts of the tensor not crucial to the problem.

Tensor Algebra: Properties of Tensors – UG 2021 Videos

Tensor Algebra: Properties of a Tensor. – UG 2025 Videos
“Continuum Mechanics may appear as a fortress surrounded by the walls of tensor notation” E. Tadmore, et. al. 1. The word “Tensor” applies to virtually all the quantities encountered in Engineering. Scalars and Vectors are zeroth and first order tensors. However, the word, with no prefix, refers to a second-order tensor. 2. For an object to be proved to be a tensor we need to show that it transforms a vector and its output is also a vector. Secondly, that transformation must be linear. 3. A tensor can be expressed in Component Form. The vector itself is more than its components as components presume reference to particular coordinate system. Outside that the numbers mean nothing. 4. For any tensor, certain scalar-valued functions are characteristic of the tensor, independent of coordinate systems. These are usually the targets of computations of any tensor. They are Principal Invariants. 5. Tensors can be decomposed additively or multiplicatively to simpler tensors. The goal is to make analysis easier and gain valuable insight by removing parts of the tensor not crucial to the problem.

Tensor Algebra: Properties of a Tensor – UG: SSG 321 2023 Videos

Week Six: Tensor: A Linear Transformation – 1 of 3

13:04
Week Thirteen: Tensor Analysis I

44:19
Week Six: Tensor: A Linear Transformation – 2 of 3

34:38
Week Six: Tensor: A Linear Transformation- 3 of 3

20:50
Week Seven: Properties of Tensors I

50:07
Week Eight: Properties of Tensors II – 1 of 2

56:10
Week Eight: Properties of Tensors II – 2 of 2

25:52
Week Nine: Properties of Tensors III – 1 of 2

37:45
Week Nine: Properties of Tensors III – 2 of 2

25:14
Week Ten: Properties of Tensors IV – 1 of 2

53:01
Week Ten: Properties of Tensors IV – 2 of 2

27:08
Week Eleven: Properties of Tensors V

48:17
Week Eleven: Eigenvalue Question I

23:09
Week Twelve: Eigenvalue Question II- 1 of 2

36:58
Week Twelve: Eigenvalue Question II- 2 of 2

34:08
Week Thirteen: Eigenvalue Question III

32:05

Tensor Algebra: Properties of a Tensor (videos)

Tensor Analysis: Differential and Integral Calculus with Tensors
“We do not fuss over smoothness assumptions: Functions and boundaries of regions are presumed to have continuity and differentiability properties sufficient to make meaningful underlying analysis...” Morton Gurtin, et al. 1. The first chapter provided the necessary background to study tensors. Chapter 2 explored the properties of tensors. 2. This third chapter begins the applications by introducing the concept of differentiation for objects larger than scalars. 3. The gradient of scalar, vector or tensor is connected to the extension of the concept of directional derivative called the Gateaux differential. 4. The complexity in differentiation of large objects comes, not from the objects themselves, but from the domain of differentiation. Differentiation rules with respect to scalar arguments follows closely the similar laws for scalar fields. 5. Integral theorems of Stokes, Green and Gauss are introduced with computational examples. 6. Orthogonal Curvilinear systems are dealt with in the addendum. Important results and the use of Computer Algebra are given.

Tensor Analysis: Differential and Integral Calculus with Tensors – UG 2021 Videos

Tensor Analysis: Differential and Integral Calculus with Tensors – UG: SSG 321 2023 Videos
“We do not fuss over smoothness assumptions: Functions and boundaries of regions are presumed to have continuity and differentiability properties sufficient to make meaningful underlying analysis...” Morton Gurtin, et al. 1. The first chapter provided the necessary background to study tensors. Chapter 2 explored the properties of tensors. 2. This third chapter begins the applications by introducing the concept of differentiation for objects larger than scalars. 3. The gradient of scalar, vector or tensor is connected to the extension of the concept of directional derivative called the Gateaux differential. 4. The complexity in differentiation of large objects comes, not from the objects themselves, but from the domain of differentiation. Differentiation rules with respect to scalar arguments follows closely the similar laws for scalar fields. 5. Integral theorems of Stokes, Green and Gauss are introduced with computational examples. 6. Orthogonal Curvilinear systems are dealt with in the addendum. Important results and the use of Computer Algebra are given.