“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!
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!
“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 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 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: 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.