Scalar done wrong
When I was updating my very brief notes on tensors, it occurred to me to post on iMechanica a request for recommendation of textbooks on linear algebra. I was delighted to see Arash respond. I then asked for his opinion about the definition of tensor. He responded again, and we seemed to agree. Then Amit joined the discussion, and then others. That thread has become very interesting and very long.
But I have another issue with the way we use linear algebra. I wish to get your opinion. The issue is about scalars.
Vector space. First some background. A vector space involves two sets and four operations. One set is the number field F, along with two operations: addition of two elements in F gives another element in F, and multiplication of two elements in F gives another element in F. The two operations follow the usual arithmetic rules. See a formal definition of number field. For our purpose, F can be the field of real numbers. There is nothing more to it. The other set V is a set of vectors, along with two more operations: addition of two elements in V gives another element in V, and multiplication of an element in F and an element in V gives an element in V. The two operations follow the usual rules for vectors. See a formal definition of vector space.
A space V is said to be n-dimensional if there exist n linearly independent elements in V, but every n + 1 elements of the space are linearly dependent.
In particular, for a one-dimensional vector space S on a number field F, any two elements in S are linearly dependent. Let u be an arbitrary, but none-zero, element in S. All other elements in S takes the form au where a is an element in F.
Is scalar a synonym of number? In many textbooks on linear algebra, the word scalar is a synonym to the word number, an element in the field F. The word scalar is also commonly used in physics to indicate quantities like mass, energy and entropy. The two usages of the word scalar are incompatible in several ways. First, a physical property like mass is more than just a number; it has a unit. Second, the multiplication defined on a field makes no sense to mass: the multiplication of two elements in F gives yet another element in F, but the multiplication of two masses does not give another mass. Third, if we regard both mass and entropy as elements in the field F, then we need to assign a meaning to the addition of mass and entropy. What does that even mean?
Thus, we will call an element in the field F a number, and will reserve the word scalar for physical quantities.
The set of pieces of a substance of all sizes. Given a substance (e.g., gold), pieces of all different amounts of the substance form a set. We can define the addition of the pieces, but we do not have a sensible definition for the multiplication of the pieces. Thus, this set is not a number field. This set, however, is a one-dimensional vector space. We stipulate the two operations in a natural way. The addition of two pieces of the substance is another piece of the substance, and multiplication of a real number and a piece of the substance is another piece of the substance.
Extensive property of a substance. A piece of a substance has many physical properties, such as volume, shape, color, temperature, mass, energy, entropy. A physical property is extensive if it is proportional to the amount of the substance. Volume, mass, energy, and entropy are extensive properties. Shape, color, temperature are not extensive properties.
Extensive scalar. We can use a one-dimensional vector space S to model a physical property such as mass. In this model, F is the field of real numbers. We call this one-dimensional vector space a scalar set. We use a particular element in S as the unit for this quantity. For example, for the scalar set S of all masses, the unit mass, kg, is a block of metal located in Sevres, France. All other masses equal this unit times a real number.
By contrast, temperature cannot be represented as a one-dimensional vector space. The addition of two temperatures does not give another temperature.
Linear form. In textbooks of linear algebra, a linear form is commonly defined as follows. Let V be a vector space on a number field F. A linear form is a linear map that maps an element in V to an element in F.
I believe that this definition is inconsistent with how mechanicians use linear form.
Here is my modified definition. Let V be a vector space and S be a scalar set, both on the same number field F. A linear form is a linear map that maps an element in V to an element in S.
Dual space. The set of all linear forms from V to S is also a vector space. We denote this space by V', and call it the dual space of V with respect to the scalar set S. The vector space V and its dual space V' have the same dimensions.
Work, displacement, and force. Here is an example how linear form arises in mechanics. Work cannot form a number field: the multiplication of two amounts of work does not give another work. Work, however, is a scalar. If the displacement is small, we know the work is linear in the displacement. That is, there exists a linear map that maps the displacement to work. The displacement is an element of a three-dimensional vector space U, and the work is an element of a scalar set. The linear map fits the definition of the linear form, and we call the linear map the force. The force is an element of the dual space U' with respect to the scalar set of work.
Tensor. We have had a long thread on the definition of tensor. Here is a definition that I like. Let V1, V2..., Vp and V be vector spaces on the same field F. These vector spaces may represent objects of different kinds, and may even have different dimensions. A tensor is a multilinear map that maps an element in V1, an element in V2,..., and an element in Vp to an element in V.
Remark 1. Tensor is a generalization of linear form.
Remark 2. The set all multilinear maps from V1, V2..., Vp to V is a vector space. This vector space generalizes dual space.
Remark 3. We can use this new vector space to generate other tensors. To be consistent, a vector is a special case of tensor, so is a scalar. They live in different vector spaces.
This is the main theme of linear algebra: vector spaces, and linear maps between them. Linear maps themseves form new vector spaces. We map the maps. The never-ending strory of linear algebra.