Introduction to Index Notation — Vectors, Basis Vectors & Einstein Summation

Based on the supplied transcript. Punctuation and paragraph breaks have been added, with repeated fillers and false starts reduced for readability. Clearly identifiable mathematical expressions have been typeset. The speaker’s commentary and opinions are retained; mathematical clarification appears separately in the AI summary.

All right, kids. We’re in the engineering month, November 2026.

From the Gustafson book, we already worked through a matrix-exponential relation and showed how it worked.

That work remains open for anyone who wants to correct the typo. The negative one should have been here. The arithmetic needs correction, and you can type LaTeX on the website and communicate with each other.

The solution approach is correct; the typo needs fixing. I’m leaving that for you because this is an interactive program.

I did the first session live on YouTube. Nobody noticed the mistake in the two-by-two adjugate, even someone who said they had studied advanced linear algebra.

That makes me question their preparation. But checking work together is part of the research program.

I’m looking for people genuinely interested in mathematics, physics, and engineering who want to participate.

You can correct the matrix and substitute it back into the exponential calculation. Sorry—I just had a spicy burrito.

For the foreseeable future, we’ll work with Mary Boas’s book and look at tensor analysis.

Before tensor analysis, we need index notation. Many students are focused on the satisfaction of solving puzzles rather than understanding the mathematical framework.

Before we start, what is a tensor?

If you don’t know, the correct response is, “I don’t know,” rather than trying to describe something from a video.

There’s a definition, different ranks, and a context. Guessing without identifying those things isn’t what we want in scientific work.

I remember a professor asking why one times zero equals zero. I said, “I have no idea.” He told me that was the right response.

I studied with people who did remarkable work, and I’m trying to carry that mentorship forward.

If you’re serious about this, I’ll make introductory videos before the live sessions where we investigate problems and applications.

We’ll begin with Mary Boas, but first we need an introduction to index notation.

An accessible starting point is a sequence. Consider terms $a_1,a_2,\ldots$. We can refer to a general term as $a_i$.

In this example, the indexing starts at one. Depending on the book’s convention, the natural numbers may mean the positive integers, or they may also include zero.

We could instead start at zero. The important thing is identifying the index set.

For a sum of the terms, we have:

We aren’t necessarily going to infinity. The upper endpoint is $j$, whatever positive integer we specify.

That’s a basic connection to what we’re studying. Now let’s move into vector analysis.

I’ll use $\mathbf{v}$ for a column vector. The terminology and notation depend on the book and subject.

We could begin with components $v_1,v_2,v_3,\ldots$, but because this journal is focusing on engineering, we’ll stay in three dimensions for now.

There are several ways to write the components of the same vector. We’re considering notation from different courses and textbooks.

Depending on whether you’re in introductory physics, calculus, quantum mechanics, or linear algebra, the presentation can change.

It’s important to understand the conventions used in the assigned text, rather than assume that every expression has the same meaning in every setting.

I compare this with APA versus MLA. There are conventions to follow.

Let’s look back through Mary Boas’s book to see how vectors are written there.

Here’s a vector expressed using $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ with the components $x$, $y$, and $z$.

The scalar may appear before or after the basis vector. For these real scalar multiples, the placement doesn’t change the result.

I need to adjust my chair and get a cushion. Sitting for this work can be uncomfortable.

I prefer working while reclining, though that might look strange on camera. I may do it during live sessions anyway.

This vector notation also appears in Thomas’ and Stewart’s calculus books. They use more than one notation.

Our main concern now is indexing. Let’s index the three-dimensional vector.

I’ll use column-vector notation. We need to understand the notation before moving into tensor definitions and operations.

We separate the vector into three pieces, with zeros in the other component positions, then factor out the scalar components.

The familiar basis vectors are $\mathbf{i}=(1,0,0)$, $\mathbf{j}=(0,1,0)$, and $\mathbf{k}=(0,0,1)$.

What if we have more components or more dimensions than we have convenient letters? We index.

We’re staying in three dimensions here, which is useful for the engineering and physics examples.

Let the unit basis vectors be represented by $\mathbf{e}_1$, $\mathbf{e}_2$, and $\mathbf{e}_3$.

Then the vector becomes the sum from $i=1$ to $3$ of $v_i\mathbf{e}_i$.

When you see the indices in the textbook, this is the kind of summation they’re expressing.

There’s also the Einstein summation convention. Under that convention, there’s no need to write the capital sigma explicitly.

Is “capital” the spelling for letters? I’m having a moment. Capital money, capital letters, Capitol building—I live near the Capitol.

We can write the vector using the repeated index, with summation understood.

That’s the introduction. Next, we’ll look at products, including indexing a dot product.

If I were teaching this as an introductory class, I’d demonstrate the vector and then give you ten minutes to try writing the dot product with indices.

We’ll use the term “dot product” here. Dot and cross products appear together in three-dimensional vector analysis.

Dot products extend to other dimensions. The cross product is associated with three dimensions and, in another setting, seven dimensions.

Infinite-dimensional indexing is a different matter. You need to know the space you’re working in.

Are we in three-dimensional space or considering something larger? How big is the computer we live in? Does God know we exist, or are we something disrupting it from within?

Anyway, this is a basic introduction to expressing $\mathbf{v}$ using indexed components and basis vectors.

For now, $i$ runs from one to three. Once we’ve established that we’re working in $\mathbb{R}^3$, we don’t need to repeat that range every time.

This is the first introduction to index notation for the engineering work in Math Monthly.

Over the years, I’ve seen physics and engineering students struggle at this point because they haven’t spent enough time understanding notation.

My view is that engineering applies physics, physics applies mathematics, and mathematics applies language.

You need to write. Look at the document I’ve typed. Microsoft gives it a high writing score.

You’ve had years of schooling and access to computers. If you can’t produce clear writing, I think that reflects a serious problem in education.

It’s also your responsibility to take initiative.

Engineers and physicists write professional documents. You need to be prepared for that part of the work.

You spend years writing English and mathematical explanations, but many students focus only on arithmetic.

I think that’s connected with the attrition and employment difficulties I’ve seen. Students reach senior year and find that far fewer of their original classmates remain.

If you’re watching early in your education, take the opportunity to improve your writing and source use now.

Or join us and participate. I expect a substantial portion of this journal to involve index notation and tensor analysis.

We do it together as a team. If you enjoy mathematics, physics, and engineering, it can be a lot of fun.

I do this every day. I’m an author; I write. I enjoy mathematics, physics, and challenging problems, especially with people interested in engaging seriously.

I’ll see you in the next one.

Author Jonathan David introduces indexing through familiar sequences and sums, then uses the standard basis of three-dimensional space to explain component notation and Einstein summation. The lesson prepares for later work on vector products and tensor analysis.

An Index Labels a Component or Term

In a sequence, $a_i$ denotes the term labeled by $i$. The index set must be specified: a sequence might begin at one or zero, depending on the context and the author’s conventions.

Here, $i$ is the summation index and $j$ specifies the final term.

Write the Vector in a Chosen Basis

For the standard Cartesian basis of $\mathbb{R}^3$:

A vector with components $v_1,v_2,v_3$ can then be written:

Adopt the Einstein Summation Convention

In the Cartesian convention used here, an index appearing twice in a term is summed over its established range. Thus:

The index $i$ is a dummy index: changing both occurrences consistently to another available label does not change the vector.

Keep the Notational Context Clear

The lesson compares component lists, column vectors, and basis-vector expressions. Their relationship depends on the chosen basis and stated conventions. A tuple should not automatically be treated as a row or column matrix without specifying how it is being represented.

Preparation for Tensor Analysis

Jonathan presents this notation as groundwork for indexed dot products, cross products, and tensor expressions. He emphasizes understanding the source’s definitions and conventions before manipulating its formulas.

Mathematical Communication

Alongside the notation, the discussion emphasizes clear technical writing and collaborative checking. Jonathan also revisits an earlier matrix calculation and invites participants to correct its arithmetic as part of the interactive journal.