Understanding Tensors: Definition and Tensor Products

1. Tensors as Multilinear Maps

The core idea is to generalize the behavior of vectors and covectors. A covector $\omega$ is a linear map that takes one vector $X$ and produces a number, $\omega (X)\in \mathbb{R}$. In a finite-dimensional space, we can also view a vector $X$ as a linear map that takes one covector $\omega$ and produces the same number, $X(\omega )=\omega (X)$.

A general tensor is simply a machine that takes multiple vectors and covectors as inputs and produces a single real number, and it must be linear in each of its input slots.

Definition: Type (r,s) Tensor

A tensor $T$ of type (or rank) (r,s) at a point $p\in M$ is a multilinear map that takes $r$ covectors (from $T_p^{\ast }M$) and $s$ vectors (from $T_{p}M$) as arguments and returns a real number.
$T:\underset{r\text{ times}}{\underbrace{T_p^{\ast }M\times \cdots \times T_p^{\ast }M}}\times \underset{s\text{ times}}{\underbrace{T_{p}M\times \cdots \times T_{p}M}}\to \mathbb{R}$
"Multilinearity" means that the map is linear in each of its $r+s$ arguments separately. For example, for a type (1,1) tensor $T(\omega ,X)$, linearity in the first slot means:
$T(a{\omega }_1+b{\omega }_2,X)=aT({\omega }_1,X)+bT({\omega }_2,X)$
and linearity in the second slot means:
$T(\omega ,c{X}_1+d{X}_2)=cT(\omega ,X_1)+dT(\omega ,X_2)$
for any scalars $a,b,c,d$, vectors $X,X_1,X_2$, and covectors $\omega ,{\omega }_1,{\omega }_2$.

This definition consistently frames our existing objects:

• A vector is a type (1,0) tensor: it takes one covector to produce a number.
• A covector is a type (0,1) tensor: it takes one vector to produce a number.
• A scalar is a type (0,0) tensor: it takes zero arguments and "returns" a number (itself).

The following lecture segment formally introduces this definition. The lecturer uses the term "(r,s) tensor" in the same way we have defined it.

This definition is also clearly stated in the notes from MIT.

2. The Tensor Product: Building Tensors

The definition of a tensor as a multilinear map tells us what a tensor does, but not how to build one. The fundamental construction tool is the tensor product (also called the outer product), denoted by the symbol $\otimes$.

The tensor product combines two tensors to create a new, higher-rank tensor. If $T$ is a type $(r_1,s_1)$ tensor and $S$ is a type $(r_2,s_2)$ tensor, their tensor product $T\otimes S$ is a type $(r_1+r_2,s_1+s_2)$ tensor.

Its action is defined in the most natural way possible: you feed the first set of arguments to $T$, the next set of arguments to $S$, and multiply the resulting real numbers.

Let's see this with the simplest non-trivial example. Let $X$ and $Y$ be two vectors (type (1,0) tensors). Their tensor product $X\otimes Y$ is a type (2,0) tensor. According to the definition, it must be a machine that eats two covectors, say $\omega$ and $\sigma$, and returns a number. The rule is:
$(X\otimes Y)(\omega ,\sigma ):=X(\omega )Y(\sigma )$
Note that the tensor product is not commutative: in general, $X\otimes Y\ne Y\otimes X$, because $(Y\otimes X)(\omega ,\sigma )=Y(\omega )X(\sigma )$, which is a different value.

This construction is the key to everything that follows. We can build any tensor we need by taking tensor products of our fundamental building blocks—vectors and covectors—and then taking linear combinations of the results.

The following video explains the rules of the tensor product and the vector spaces they form.

3. Tensor Spaces and Components

The set of all type (r,s) tensors at a point $p$ forms a vector space, which we denote as $T_p^{(r,s)}(M)$. This vector space is itself the tensor product of the underlying tangent and cotangent spaces:
$T_p^{(r,s)}(M)=\underset{r\text{ times}}{\underbrace{T_{p}M\otimes \cdots \otimes T_{p}M}}\otimes \underset{s\text{ times}}{\underbrace{T_p^{\ast }M\otimes \cdots \otimes T_p^{\ast }M}}$
This is the formal, abstract definition of the space of tensors. Given your preference for mathematical rigor, this is the proper starting point. The identification of an element of this space with a multilinear map is a consequence of a deep result in linear algebra called the universal property of the tensor product.

This property guarantees a one-to-one correspondence between:

1. Elements of the abstract tensor product space $T_p^{(r,s)}(M)$.
2. The multilinear maps we defined in Section 1.

This assures us that both ways of looking at tensors are equivalent.

Tensor Components

Just as we can expand a vector $X$ in a basis $X=X^{\mu }{\partial }_{\mu }$, we can expand any tensor in a basis constructed from the bases of $T_{p}M$ and $T_p^{\ast }M$.

A basis for the space of type (r,s) tensors, $T_p^{(r,s)}(M)$, is formed by all possible tensor products of the basis vectors $\{{\partial }_{\mu }\}$ and basis covectors $\{d{x}^{\nu }\}$. For a type (1,1) tensor, the basis would be the set of $n^2$ objects $\{{\partial }_{\mu }\otimes d{x}^{\nu }\}$.

An arbitrary type (1,1) tensor $T$ can therefore be expanded in this basis:
$T=T^{\mu }{}_{\nu }({\partial }_{\mu }\otimes d{x}^{\nu })$
The $n^2$ numbers $T^{\mu }{}_{\nu }$ are the components of the tensor $T$ in this coordinate basis.

How do we find these components? We "feed" the tensor the basis vectors and covectors.
$T^{\mu }{}_{\nu }=T(d{x}^{\mu },{\partial }_{\nu })$
This is the general procedure: to find the component of a tensor, you evaluate the tensor map on the corresponding basis vectors and covectors.

For a highly rigorous and abstract construction of the tensor product space and its properties, the following resource is excellent. It uses a quotient space construction that is common in pure mathematics.

Conclusion

In this lesson, we have erected the complete scaffolding for tensor algebra. We now have a machine that can produce objects of arbitrary complexity to describe physical quantities in spacetime.

Key Takeaways:

• A tensor of type (or rank) (r,s) is a multilinear map that takes $r$ covectors and $s$ vectors as input and produces a real number.
• Vectors are type (1,0) tensors, and covectors are type (0,1) tensors.
• The tensor product $\otimes$ is the primary tool for building higher-rank tensors from lower-rank ones. For example, $(X\otimes \omega )(\sigma ,Y)=X(\sigma )\omega (Y)$.
• The set of all type (r,s) tensors forms a vector space, denoted $T_p^{(r,s)}(M)$, which can be formally defined as the tensor product of the tangent and cotangent spaces.
• In a given basis, a tensor can be represented by a multi-dimensional array of numbers called its components. For a type (r,s) tensor $T$, its components are found by evaluating it on the basis vectors and covectors: $T^{{\mu }_1\dots {\mu }_r}{}_{{\nu }_1\dots {\nu }_s}=T(d{x}^{{\mu }_1},\dots ,d{x}^{{\mu }_r},{\partial }_{{\nu }_1},\dots ,{\partial }_{{\nu }_s})$.

We have defined tensors, but the spaces $T_{p}M$ and $T_p^{\ast }M$ are so far just vector spaces. There is no notion of length, angle, or distance. This is like doing geometry without a ruler. In our next lesson, we will introduce the most important tensor in General Relativity: the metric tensor, $g_{\mu \nu }$. This (0,2) tensor is the tool that endows spacetime with a geometric structure, allowing us to measure lengths of vectors and angles between them. It is the central object in the Einstein Field Equations.