Finally.

a course that actually explains

First things first:

If you’re confused about tensors, it’s not your fault.

The usual physics curriculum does a terrible job introducing students to tensors—despite the fact that they’re some of the most important mathematical objects in physics.

You might have run into a few examples here and there—like the rotational inertia tensor $I_{ij}$ in a Newtonian mechanics class.

But most students don’t get a proper introduction to tensors until it’s jam-packed into the first two weeks of an advanced class on general relativity—Einstein's theory of gravity in curved spacetime.

That’s NUTS.

Tensors are already a challenging-enough subject.

Expecting students to learn them at the same time that they’re thrown into Einstein's mind-bending world of curved spacetime is downright cruel.

You don’t need to know anything about general relativity to understand tensors.

You can (and should) learn about tensors starting from the regular old world of Newtonian mechanics—and gradually work your way up from there.

Wouldn’t it be nice if there were a course about tensors aimed at intermediate physics students that meets you where you’re at?

A course that guides you through the fundamentals of tensors, starting from the language of things you already know, like vectors in 3D space?

That's the course I set out to create:

A clear and accessible introduction to tensors that any sophomore-level physics student can understand from start to finish.

Tensors are essential tools for understanding a huge range of subjects in physics.

Whatever physics topic you're interested in, whether it's

• Newtonian mechanics
• Electromagnetism
• Relativity
• Quantum field theory

and many more, you're going to be running into tensors all over the place.

And that means you need to know how to work with them. 

The basic reason that tensors are so important is simple:

Tensors are the geometric objects that let us write down the laws of physics as coordinate-independent equations.

And by the way, you're already very familiar with one tensor equation: $\vec F = m \vec a!$

That's because $\vec F$ and $\vec a$ are vectors—and vectors are just the simplest examples of tensors.

But crucially, vectors alone are not enough. We need tensors, too!

And you'll learn many physical examples of tensors in this course, including

• the rotational inertia, stress, and strain tensors in Newtonian mechanics
• the field strength, polarizability, and conductivity tensors in electromagnetism
• the metric, stress-energy, and Riemann curvature tensors in relativity

But before you can get to the physics applications, you'll need to understand how tensors are even defined in the first place.

And, unfortunately, textbooks tend to make that much harder than it needs to be…

There are two common definitions of tensors out there that you may have run into in physics and math textbooks.

And they both kind of suck for physics students trying to learn about tensors for the first time.

The definition you'll find in physics books goes something like this:

"A rank $\boldsymbol{(n,m)}$ tensor $\boldsymbol{T^{i_1\cdots i_n}{}_{j_1\cdots j_m}}$ is a set of numbers labeled by $\boldsymbol{n}$ upper indices and $\boldsymbol{m}$ lower indices that transforms according to the rule

$$\begin{align}
\boldsymbol{T}&^{\boldsymbol{i_1'\cdots i_n'}}\boldsymbol{{}_{j_1' \cdots j_m'}\vphantom{\frac{\partial  x^{i_1'}}{\partial x^{i_1} }}}\\
=& \boldsymbol{\frac{\partial  x^{i_1'}}{\partial x^{i_1} }\cdots \frac{\partial x^{i_n'} }{\partial x^{i_n} } 
\frac{\partial  x^{j_1}}{\partial x^{j_1'} }\cdots \frac{\partial  x^{j_m}}{\partial x^{j_m'} }}\\
&\times \boldsymbol{T^{i_1\cdots i_n}{}_{j_1\cdots j_m}}
\end{align}$$

under a coordinate transformation from coordinates $\boldsymbol{x}$ to $\boldsymbol{x'}.$"

Yikes.

I don't know about you, but I can barely make it halfway through that definition before my eyes start to glaze over 😵‍💫. 

Is it any wonder students come away thinking tensors are scary?!

You will understand what this complicated looking equation means in this course, but we're going to work our way up to it gradually.

In math books, meanwhile, you'll see a very different looking definition of tensors:

This version doesn't have any complicated equations or indices in it, at least!

And yet, it must still seem very mysterious for a physics student encountering it for the first time.

"Multi-linear maps?"

"Dual vectors?"

"What does all that abstract math have to do with the physics objects I'm used to working with?"

You'll understand both of these definitions—and how they're related to each other—by the end of this course.

But I'm not going to start the course with either of them.

Instead, I'm going to show you that tensors are straightforward extensions of the things you already understand: vectors in 3D space.

And once you've made the leap from vectors to tensors, you'll unlock a whole new perspective on some of the deepest topics in physics—like the tensor formulation of electromagnetism.

Depending on how much electricity and magnetism you've studied so far, you may have run into Maxwell's equations:

$$\begin{align}&\vec\nabla \cdot \vec E = \rho \vphantom{\frac{\partial  \vec B}{\partial t }}\\ &\vec\nabla \cdot \vec B = 0\\
&\vec\nabla \times \vec E = - \frac{\partial \vec B }{\partial t }\\
&\vec\nabla \times \vec B = \vec J +\frac{\partial  \vec E}{\partial t } \end{align}$$

Essentially, they're the analog of $\vec F = m \vec a$ for electric and magnetic fields.

Their discovery was the crowning achievement of 19th-century physics.

And yet… they're kind of hideous.

I mean, just look at them. 

They're a jumbled mess of three different kinds of derivatives:

• the divergence $\vec\nabla \cdot \vec V$
• the curl $\vec\nabla \times \vec V$
• and time derivatives $\partial \vec V/\partial t$

Not to mention the fact that there are four of them.

So why do Maxwell's equations come out looking so complicated, whereas $\vec F = m \vec a$ is beautifully simple?

Because we're writing them in a dumb notation!

Or at least, a notation that hides the true mathematical structure of electromagnetism.

You see, $\vec E$ and $\vec B$ are actually just pieces of a more fundamental object: the electromagnetic tensor $F_{\mu\nu}.$

And in the language of tensors, Maxwell's equations boil down to essentially one line:

$$\nabla_\nu F^{\mu\nu} = J^\mu.$$

Now that's a beautiful equation!

To understand it, though, you'll need to learn the tensor index notation that's used all throughout higher-level physics.

Crack open most any advanced physics book and you're likely to be bombarded with tensor equations.

Equations that often come packed with indices.

Take Einstein's famous $E = mc^2,$ for example—or more generally, $E = \sqrt{(mc^2)^2 + (\boldsymbol p c)^2}$ for the energy of a particle with momentum $\boldsymbol p.$

It's an equation that comes up constantly—and you'll find it written in tensor notation in at least half a dozen different ways:

$$\begin{align}-m^2c^2 =~& p_\mu p^\mu \vphantom{\sum}\\ =~& p^\mu p_\mu \vphantom{\sum}\\ =~& \eta_{\mu\nu}p^\mu p^\nu \vphantom{\sum}\\ =~& \eta^{\mu\nu}p_\mu p_\nu \vphantom{\sum}\\ =~& -(p^0)^2 + p^i p_i \vphantom{\sum}\\ =~&\vec p\cdot \vec p.\end{align}$$

And this is a simple example of an index equation.

Learning to deal with indices can be a major roadblock for students trying to break into higher-level physics.

It's literally written in a completely different language from what you're used to!

And you might be left wondering…

• … why are some indices like $p^\mu$ written as superscripts, while others like $p_\mu$ are subscripts?
• … why are Greek letters like $\mu$ and $\nu$ used in some places, while regular letters like $i$ and $j$ are used in others?
• … what does it mean when the same index is repeated, like in $p^\mu p_\mu?$

As with any new language, tensor index notation takes time and practice to become fluent.

And by the end of this course, you'll be able to read, write, and manipulate equations like these with ease.

At the same time, though, it's easy to get lost in the sea of indices and forget what the equations actually mean geometrically. 

That's why I'm going to teach you both the practical skills for manipulating indices and the geometry behind them.

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An online course has A LOT of advantages over a traditional one.

📅 You can complete the lessons on your own schedule.

🏠 You can attend from the comfort of your own home.

⏯️ You can pause, rewind, and rewatch.

(I can't tell you the number of times I'd wished I had a remote control to pause or rewind a professor's lecture during classes I took in college!)

There’s one major disadvantage, though.

And it's that I’m not there with you in class to answer your questions.

That’s why I’m offering an optional add-on for 1-to-1 Coaching Support:

Due to the number of students taking the class at once, it unfortunately won't be possible for me to give detailed question and answer support to everyone.

(Though I do my best to answer quick questions in the course comment sections!)

But with the 1-to-1 Coaching add-on, I’ll be available to answer your questions as you progress through the course.

Any time you…

🙋 get stuck on a problem…

🙋 get confused about an equation…

🙋 need help understanding any concepts from the course…

you'll be able to send me a message, and I'll help as soon as I can.

I can only provide 1-to-1 Coaching Support to a limited number of students at a time, however, so please be aware that this offer may sell out.

If you're not sure yet if the course is right for you, take a look at the FAQs below, and also feel free to write to me at [email protected] with any additional questions.

If you have any additional questions about whether the course is right for you, feel free to write to me at [email protected].

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