Why Index Notation Comes First — Preparing for Tensor Analysis

Based on the supplied transcript. Punctuation and paragraph breaks have been added, with repeated fillers and false starts reduced for readability. Obvious name transcriptions have been corrected. The speaker’s recollections, opinions, and tentative mathematical statements are retained; technical clarification appears separately in the AI summary.

Kids, the topic for the foreseeable future is index notation and tensor analysis for the engineering book—the first official journal we’re working on together.

I did this with a group over the summer. We called it PLEM Academy, but I was busy with weekly fiction book signings. I was stressed, trying to do both, and things became jumbled.

I’m here to help mathematics, physics, and engineering students who understand how to read and write, or want to develop that ability.

In my view, an engineer is an applied physicist, a physicist is an applied mathematician, and a mathematician is an applied linguist. An engineer draws on all of those.

You may imagine working in Iron Man’s laboratory, tinkering. Some jobs involve building and assembling things, but afterward you’ll need to write documents explaining the process so others can reproduce it.

Whether you’re building something or not, professional writing matters. That’s a component of the research program.

I’m a professional author. I’ve sold fiction, children’s books, cookbooks, and mathematics and physics books. I’m now focusing on mathematics and physics.

During the pandemic, long COVID kept me from doing mathematics for about 18 months, so I focused on fiction. That experience helped me realize I didn’t want to remain in entertainment.

I struggled with the people and communication in that world. Then, coming to YouTube to discuss science, I saw problems there too.

Students have spent years in school but struggle to edit a sentence while preparing for engineering.

If you don’t want to spend substantial time studying mathematics and thinking about it outside study sessions, why choose this degree?

It’s like my dog trying to attack a fire truck. If she hadn’t been on a leash, it could have ended badly. She has a lot of instinct, and I have to be careful downtown.

For the foreseeable future, we’ll use Mary Boas’s book and perhaps other sources to study index notation and tensor analysis.

This is where Einstein’s mathematical difficulties become part of the story.

Some students say they can be physicists while being bad at mathematics because Einstein was supposedly bad at it. The difficulties I’m talking about involve advanced theoretical mathematics, not basic writing or algebra.

Tensor analysis is complicated and takes time. I think modern typing and copy-and-paste tools make working through its calculations easier.

Showing how a cross product becomes a compact indexed expression can require pages of intermediate work.

The compact expression then lets you perform calculations without repeatedly writing all that background.

My interpretation is that Einstein had difficulty with some of the underlying pure mathematics. I’m basing that partly on what I’ve read of his exchanges with Levi-Civita.

I find those letters entertaining, like a modern string of text messages. I’m exaggerating theatrically: Einstein asserts his position, Levi-Civita challenges the mathematics, and eventually Einstein accepts the correction.

It reminds me of arguments online where someone wants to use mathematics without understanding the framework.

We’re starting at the beginning. I just made a video showing how to index a vector and use Einstein summation convention.

That becomes a rank-one tensor—or rank two, depending on the book. I think in Mary Boas, rank one is a Cartesian tensor, rank two a vector, and rank three a matrix. I need to check and stay in sync with her definitions.

You can’t identify something just by looking at a symbol without knowing the context.

In an electrodynamics discussion, I wrote an expression involving a bold capital F and other symbols and asked whether it was true.

People began discussing conservative fields. But I hadn’t defined the symbols, identified the subject, or referenced a source. The expression didn’t yet have an established meaning.

I’m testing whether you’ll admit you don’t know. You won’t get far if you can’t say that.

There’s more mathematics and physics than anyone can master. School gives you an introduction across topics, then you specialize in a discipline through work or further study.

I began in mechanical engineering and physics, then moved toward mathematics and physics.

I was approaching graduation with a double major at UT Dallas when I ran out of money.

I told my theoretical physics professor what interested me and that I disliked the laboratory courses.

He suggested pure mathematics as a route toward the theoretical work I wanted. I pursued that direction, then the pandemic and long COVID disrupted it.

It was a nightmare, but I recovered. I feel I understand mathematics better now. The time away seemed to let some connections settle.

As Forrest Gump might say, God works in mysterious ways.

For engineers and physicists interested in the research program, this month may be especially valuable.

While tutoring, I repeatedly heard from junior-year students struggling with tensor analysis. I told some that they needed to reconsider their major or work through the material independently.

I’ve seen people spend years watching others do mathematics and copying the process without developing independence.

If you aren’t willing to lift the weights, don’t go to the gym. I compare having someone do the work for you with having them lift a weight while your hands remain underneath it.

What we’re doing is collaborative exploration, not completing assignments for someone.

Sometimes I don’t know how to solve the problem until I read the source. We investigate together.

Over the summer, a student saw a connection I missed. At first, I disagreed, then realized the student was right. That helped us move forward faster than I would have alone.

That’s research together. I’m a guide, not an expert in every question we encounter.

The objective is to use unfamiliar books and questions, not simply repeat things I’ve already solved.

Tensor analysis matters in many areas of physics and engineering. It can extend beyond three dimensions, and index notation can be used in higher-dimensional settings.

I want to help students overcome the difficulty I’ve seen at this transition.

The introductory work will develop the calculations and indexing that lead into tensor notation.

Books may omit intermediate steps because the reader is expected to investigate them. When something is unfamiliar, use that as a reason to work through it rather than have someone complete it for you.

I was in those courses too. I used outside material until I read the university syllabus and code of conduct.

One syllabus warned against possessing the instructor’s solution manual and restricted resources to the textbook and lecture notes. It also prohibited working with classmates.

I accepted that challenge. That’s part of how I developed the ability to work as I do.

I know there may be a missing component in your education. You might feel criticized, but I’ve been in similar circumstances.

I’m trying to help you become your own guide. Sometimes you need someone to guide you toward independence.

The journal is approximately 200 pages. Over the next week or two, perhaps 50 to 100 pages will involve index notation leading to tensor analysis.

It may become most of the volume. Then we may investigate applications such as heat-rod problems.

Again, if I ask what a tensor is and you don’t know, say so rather than guess.

We’re working with precise mathematical definitions before discussing applications.

I’ll share a few introductory videos and snippets. The research sessions are private group meetings, currently through Zoom, rather than public YouTube streams.

It’s less expensive than the private tutoring requests I used to receive from students wanting many hours of individual help.

I knew someone who graduated with a high physics GPA after relying on solution manuals and outside assistance, then struggled to find work in the field.

A degree and GPA don’t necessarily demonstrate independent competence.

I think too many students learn to copy work rather than develop the skills an employer wants.

I’ve said that, outside certain universities, a very high GPA can make me suspicious. In my view, an average around 2.75 to 3.25, or sometimes 2.5, may be sufficient for employment when accompanied by practical abilities.

I emphasize Microsoft Office and communication skills. I’ve shown job postings and will show more.

For those following my dental situation, I postponed the implants because of an insurance issue and the possible interruption to speaking. I need to keep working for now.

That’s the plan: Engineering, November 2026, with index notation and tensor analysis.

This is the physical journal. I manufacture these books by hand. If you order the physical version, I sign it and mail it to you.

Math Monthly, that monthly monthly. Check it out. I hope to see you in the live sessions. Have a nice day—unless you have other plans.

Author Jonathan David explains why the Math Monthly engineering work will begin with index notation before moving into tensor analysis. He connects the topic with careful reading, technical communication, independent reasoning, and collaborative research.

Build the Notational Foundation

The planned progression begins with vector components and summation notation, then develops the calculations behind more compact expressions. The goal is to make tensor formulas readable rather than treat them as unexplained collections of symbols.

Clarifying Tensor Order

The recording contains a tentative, inconsistent description of tensor ranks. In the standard tensor-order terminology relevant here:

• A scalar is an order-zero tensor.
• A vector is an order-one tensor.
• An order-two tensor has two component indices and can be represented by a matrix once bases are chosen.

Tensor order is distinct from matrix rank. A matrix-shaped array also requires the appropriate transformation interpretation to represent a tensor in the intended setting.

Define Symbols Before Interpreting Them

Jonathan uses an expression with undefined symbols to illustrate why context matters. Before deciding whether a formula is true, the objects, operations, assumptions, and notation must be specified.

Work Through the Hidden Steps

A compact indexed expression may stand for a longer component calculation. Jonathan proposes working through those intermediate steps and using typed mathematical documents to organize the process.

Collaborative Checking

Jonathan describes a session in which a participant noticed a connection he had initially missed. He presents this as a reason to investigate unfamiliar problems together and remain willing to revise an interpretation.

Technical Writing & Professional Skills

The discussion emphasizes explaining procedures, preparing documents, and communicating clearly. These are presented as important complements to mathematical calculations.

Upcoming Research Work

Mary Boas’s textbook is the proposed starting point, with other sources potentially added. Jonathan anticipates developing substantial journal material on index notation and tensor analysis before exploring additional applications.

The Einstein–Levi-Civita exchange is presented in the recording as a theatrical retelling, not a verified quotation. Statements about GPA and hiring reflect the speaker’s opinions; GPA alone does not establish misconduct or determine employability.