TENSORS

The course is aimed at the level of an intermediate college physics student.

(But you certainly don't have to be in college to enroll—lifelong learners are more than welcome!)

Since tensors are generalizations of vectors, the most important prerequisite is to be comfortable with vectors and vector calculus. 

For example, you should know how to expand a vector $\vec A$ in a basis,

$$\vec A = A_x \hat x + A_y \hat y + A_z \hat z,$$

and how to pick out the components using the dot product,

$$A_x = \hat x \cdot \vec A.$$

You should also know how to take the gradient of a scalar function $f$ to get a vector:

$$\vec\nabla f = \frac{\partial  f}{\partial x }\hat x + \frac{\partial f }{\partial y }\hat y + \frac{\partial  f}{\partial z }\hat z.$$ 

You don't have to be an expert with vector calculus, though. We'll review the key concepts in the first few lessons of Part I, so it's okay if you're a bit rusty.

As for physics background, you should of course be well acquainted with $\vec F = m \vec a,$ and some familiarity with electric and magnetic fields will also come in handy.

For Part III, it will also be helpful if you've seen a little special relativity before, but again, you don't have to be an expert. 

Of course, as you’re going through the lessons, if you find that your preparation is a little weak on one or another topic, the beautiful thing about an online course like this is that you can just pause and step away to brush up on whatever you need to.

When you return, the course will be ready and waiting for you to pick right back up where you left off!

Part I: Fundamentals of Cartesian Tensors

Lesson 0: Why Tensors?

• Why tensors are the language of physics. Laws of physics as geometric equations. A better way to learn tensors. Course overview.

Lesson 1: The Geometry of Vectors

• Review of vectors. Vector addition and scalar multiplication. Coordinate independence of vectors. Cartesian coordinate systems. Coordinate basis vectors. Unit vectors. Vector components. Position vector. Orthonormal bases. Matrix notation. Dot product. Cross product.

Lesson 2: Trek Along a Trajectory

• Kinematics. Position, velocity, and acceleration. Displacement vector. $\vec F = m \vec a$ in Cartesian coordinates. Inertial reference frames. Time-independent vs time-dependent coordinate transformations. Derivative along a curve. Chain rule. Work-energy theorem. Angular momentum.

Lesson 3: Derivatives, in 3D!

• Scalar and vector fields. Directional derivative of a scalar field $\nabla_{\vec V} f.$ Differential $\mathrm d f.$ Gradient $\vec\nabla f.$ Geometric meaning of gradient. Steepest ascent. Derivatives as linear operators. Directional derivative of a vector field $\nabla_{\vec V}\vec A.$ Gradient of a vector field $\vec\nabla \vec A$ as a rank 2 tensor. Level sets. Vector chain rule. Conservative forces and potential energy, $\vec F = - \vec\nabla U.$ Electric field and electric potential, $\vec E = - \vec\nabla V.$

Lesson 4: Introducing Index Notation

• Vector operations in component notation. Introducing index notation in Cartesian coordinates. Vector components and basis vectors in index notation. Kronecker delta symbol $\delta_{ij}.$ Dummy vs free indices. Summation convention. Dot product in index notation $\vec A \cdot \vec B = A_iB_i.$ Contracted indices. Gradient, directional derivative, differential, and chain rule in index notation. Matrix multiplication in index notation. Anti-symmetric symbol $\epsilon_{ijk}.$ Cross product in index notation $\vec A \times \vec B = \epsilon_{ijk}A_jB_k\hat e_i.$ Epsilon identities.

Lesson 5: Rotate Your Perspective

• Coordinate and basis independence of vectors. Change of basis. Translations, reflections, and rotations of Cartesian coordinate systems. Rotation matrices and the special orthogonal group $\mathrm{SO}(3).$ Coordinate rotations $x_i' = (\mathsf R)_{ij} x_j.$ Vector component transformation law $A_i' = (\mathsf R)_{ij} A_j.$ Invariance of the dot product. Galilean group.

Lesson 6: From Vectors to Tensors

• Why vectors are not enough. Physical motivation for tensors. Overview of tensors. Tensor product. Rank 2 tensors. Basis expansion. Euclidean metric tensor. Contractions of rank 2 tensors. Components of a rank 2 tensor. Trace of a rank 2 tensor. Matrix notation for rank 2 tensors.

Lesson 7: The Rotational Inertia Tensor

• Rigid body rotations. Angular momentum and angular velocity. Rotational inertia tensor $\boldsymbol I = \sum m(r^2 \boldsymbol g - \vec r \otimes \vec r).$ Geometric vs component form of tensor equations. Deriving $\vec L = \boldsymbol I\cdot \vec \omega.$ Principal axes of a rigid body. Rotational kinetic energy $K = \frac{1}{2} \vec\omega \cdot \boldsymbol I \cdot \vec \omega.$

Lesson 8: Tensors, Mathematically Speaking

• Vector spaces. Basis vectors. Tensor product of vector spaces. Basis for tensor product space. Tangent space $\mathrm T_p.$ Tangent vectors. Tensors as elements of $\mathrm T_p{}^{\otimes n}.$ Basis expansion of a general tensor. Components of a rank $n$ tensor. Tensor fields. Comparing definitions of tensors. Physics applications. Stress tensor $\mathrm d \vec F = \boldsymbol T \cdot \mathrm d \vec A.$ Pressures and shears. Static fluid stress tensor field $\boldsymbol T = p \boldsymbol g.$ Euler's equation $\rho \vec a = - \vec\nabla p + \rho \vec g.$ Lorentz force law $\vec F = q(\vec E + \vec v \times \vec B).$ Current density vector $\vec J.$ Conductivity tensor $\boldsymbol\sigma.$ Ohm's law $\vec J = \boldsymbol \sigma \cdot \vec E.$ Drude model. The classical Hall effect.

Lesson 9: Tensor Algebra

• Contractions of rank $n$ tensors. Component form of contractions. Dot product of rank $n$ tensors. Cartesian components of a rank $n$ tensor. Symmetric and anti-symmetric parts of tensors. Anti-symmetric (Levi-Civita) tensor $\boldsymbol \epsilon.$ Geometric meaning of $\boldsymbol \epsilon.$ Dual tensor $\star \boldsymbol T.$ Cross product in higher dimensional spaces.

Lesson 10: Tensor Calculus

• Higher rank tensor fields. Directional derivative of a tensor field $\nabla_{\vec V}\boldsymbol T.$ Gradient of a tensor field $\vec\nabla \boldsymbol T.$ Divergence, curl, and Laplacian as tensor operations. Derivative of a tensor field along a curve. Maxwell's equations. Poisson's equation. Electric flux. Conservation of charge $\frac{\partial \rho}{\partial t } + \vec\nabla \cdot \vec J = 0.$

Lesson 11: Rotations, Revisited

• Primed index notation. Vector transformation law $A_{i'} = (\mathsf R)_{i'i} A_i.$ Tensor transformation law $T_{i'j'} = (\mathsf R)_{i'i} (\mathsf R)_{j'j}\, T_{ij}.$ Geometric vs transformation law definition of a tensor. Force density $\vec f = - \vec\nabla \cdot \boldsymbol T.$ Conservation of mass $\frac{\partial \rho}{\partial t } + \vec\nabla \cdot \vec G = 0.$ Stress tensor as the flux of momentum. Stress tensor for an ideal fluid $\boldsymbol T = p\boldsymbol g + \rho \vec v \otimes \vec v.$ Conservation of momentum $\frac{\partial \vec G}{\partial t } + \vec\nabla \cdot \boldsymbol T = \vec f_\mathrm{vol}.$ Electromagnetic stress tensor
  $$\begin{align}
  \boldsymbol T_\mathrm{EM} =~~~ \epsilon_0 &\left(\frac{1}{2} |\vec E|^2 \boldsymbol g - \vec E \otimes \vec E \right)\\
  + \frac{1}{\mu_0} &\left( \frac{1}{2} |\vec B|^2 \boldsymbol g - \vec B \otimes \vec B \right).
  \end{align}$$ Conservation of electromagnetic momentum $\frac{\partial \vec G}{\partial t } + \vec\nabla \cdot \boldsymbol T_\mathrm{EM} = - \vec f.$

Part II: Fundamentals of Curvilinear Tensors

Part II is still under construction, but below is an abbreviated list of the topics I plan to cover.

Non-orthonormal basis vectors $\vec e_i \cdot \vec e_j = g_{ij}.$ Dual basis vectors $\vec e{}\,^i \cdot \vec e_j = \delta^i{}_j.$ Upper (contravariant) and lower (covariant) components $A^i \vec e_i = A_i \vec e{}\,^i.$ Index notation. Geometric meaning of lower and upper indices. Raising and lowering indices $A_i = g_{ij} A^j,$ $A^i = g^{ij} A_j.$ Metric tensor $g_{ij}.$ Line element $\mathrm{d} s^2 = g_{ij} \mathrm{d}x^i \mathrm{d}x^j.$ Coordinate systems $x^i.$ Coordinate curves. Coordinate basis vectors $\vec e_i = \partial \vec r/\partial x^i.$ Dual coordinate basis vectors $\vec e{}\,^i = \vec\nabla x^i.$ General coordinate transformations. Polar, spherical, cylindrical coordinates. Coordinate vs orthonormal basis vectors. Div, grad, curl, and Laplacian in curvilinear coordinates. Tensors in curvilinear coordinates. Tensor components $\boldsymbol T = T^{i}{}_{jk} \vec e_i \otimes \vec e{}\,^j\otimes \vec e{}\,^k.$ Raising and lowering tensor indices. Covariant derivative $\nabla_i A^j = \partial A^j/\partial x^i + \Gamma^j_{ik} A^k.$ Connection coefficients/Christoffel symbols $\Gamma^i_{jk}.$ Covariant derivative of a tensor. Computer software for tensors. Generalized tensor transformation law. Generally covariant equations. Tensor algebra. Levi-Civita tensor and volume element $\sqrt{g}~ \mathrm{d}^nx.$ $\vec F = m \vec a$ in general coordinate systems. Geodesic equation.

Part III: Fundamentals of Spacetime Tensors

Part III is still under construction, but below is an abbreviated list of the topics I plan to cover.

4D spacetime. Minkowski metric tensor $\eta_{\mu\nu}.$ Line element/invariant interval $\mathrm{d}s^2 = -c^2 \mathrm{d}t^2+\mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2.$ East coast/west coast sign conventions. Lorentz coordinates $x^\mu = (ct,x,y,z).$ 4-vectors $A^\mu.$ Timelike, spacelike, null vectors. Time-space split. Proper time $\tau.$ Worldline $x^\mu(\tau).$ Relativistic free particle. Geodesic equation in spacetime $\mathrm{d}^2 x^\mu/\mathrm{d}\tau^2 + \Gamma^\mu_{\rho\sigma} \mathrm{d}x^\rho/\mathrm{d} \tau \,\mathrm d x^\sigma/\mathrm d \tau = 0.$ 4-momentum $p^\mu.$ Mass-shell relation $p^\mu p_\mu = -m^2c^2$ and Einstein's energy equation $E = \sqrt{(mc^2)^2 + (pc)^2}.$ Lorentz transformations $x^{\mu'} = \Lambda^{\mu'}{}_\nu x^\nu.$ Lorentz tensors vs generally covariant tensors. The electromagnetic gauge potential $A_\mu.$ The electromagnetic field strength tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$ Transformation of electric and magnetic fields. Maxwell's equations in generally covariant form $\nabla_\nu F^{\mu\nu} = J^\mu.$ The Bianchi identity $\partial_{[\mu}F_{\nu\rho]} = 0.$ Gauge invariance $A_\mu \to A_\mu + \partial_\mu \phi.$ Lorentz force law $f^\mu = qF^{\mu\nu}v_\nu.$ Continuity equation $\nabla_\mu J^\mu = 0.$ Tensors as linear maps. Dual of a vector space. Tangent vectors and one-forms. Isomorphism of vectors and one-forms via the metric. Recovering the tensor transformation law. Generalization to curved spaces. Lack of Cartesian/Lorentz coordinates. Coordinate basis vectors and one-forms in curved space. Path dependence. Parallel transport. Riemann curvature tensor $R^{\mu}{}_{\nu\rho\sigma},$ Ricci tensor $R_{\mu\nu},$ Ricci scalar $R.$

I think this course will be extremely helpful if you're aiming to study general relativity in the future—which requires working with tensors in curved spacetimes.

Please note, though, that the majority of this course focuses on tensors in flat space and flat spacetime.

And that's 100% intentional. Because you can already learn the vast majority of what you need to know about tensors in the much simpler world of flat space, without having to pile on the extra confusions that come from dealing with curved spaces.

In particular, you'll learn all about index notation, the metric tensor, covariant derivatives, Christoffel symbols, the geodesic equation, general covariance, and much more—all essential tools for tackling general relativity.

At the end of Part III, I'll give a brief introduction to curved spaces and describe how what we've learned about tensors extends beyond flat space. And I'll also introduce you to the Riemann curvature tensor, which is the mathematical representation of curvature and which plays an essential role in general relativity.

No! YouTube videos are great for quickly getting a taste of a topic and hopefully inspiring your interest to learn more. But you can’t truly learn advanced math and physics by passively watching a 20-minute video. There’s no substitute for slowly and methodically going through the material and solving lots of practice problems, and that’s what we’ll do in this course.

Part I is available as a separate purchase at the link below. Part II and Part III are still under construction.

Part 1

As soon as you complete your order, you'll have instant access to all of Part I.

I'm still creating Part II and Part III, and I'll gradually be releasing those lessons over time. Students who have pre-ordered the full series will be the first to have access to those lessons.

Your access does not expire—in fact, I encourage you to revisit the course a few months after you first take it to really solidify your understanding. That's one of the huge advantages of an online course like this.

I devote time every week to answering questions in the course comment sections.

However, due to the volume of students taking my courses, it isn't always possible for me to provide detailed question and answer support to everyone. That's why I'm offering the 1-to-1 Coaching option for priority support for a limited number of students.

If you sign up for 1-to-1 Coaching, I'll provide priority email support to answer your questions about the course material. Just send me a message any time you have a question, and I’ll typically respond within 24-48 hours.

These benefits will last until 4 months after all the course materials have been posted, to ensure you have time to complete the course and ask all your questions.

Part I: Fundamentals of Cartesian Tensors

• Lesson videos (~14 hours)
• Practice problem sheet and solution PDFs (300+ pages)
• Lecture summary sheet PDFs (~80 pages)
• Lecture slide PDFs (450+ pages)
• Formula sheet PDF 

Part II: Fundamentals of Curvilinear Tensors

Coming soon!

• Lesson videos
• Practice problem sheet and solution PDFs
• Lecture summary sheet PDFs
• Lecture slide PDFs
• Formula sheet PDF 

Part III: Fundamentals of Spacetime Tensors

Coming soon!

• Lesson videos
• Practice problem sheet and solution PDFs
• Lecture summary sheet PDFs
• Lecture slide PDFs
• Formula sheet PDF 

Part I: Fundamentals of Cartesian Tensors

Part I consists of about 14 hours of video lessons, plus 10 in-depth problem sheets for hands-on practice. 

At the pace of an intensive college class, it would take at least 6 weeks to complete. But I encourage you to take it at a more comfortable pace of at least 10-12 weeks to give yourself more time to absorb the material.

The pace is completely up to you, however. So if you're fitting the course into your schedule next to a full-time job or other classes, you'll be able to complete the lessons on your own time-table.

Part II: Fundamentals of Curvilinear Tensors

Part II is still under construction, but I expect the duration to be similar to Part I.

Part III: Fundamentals of Spacetime Tensors

Part III is still under construction, but I expect the duration to be similar to Part I.

The lessons don't have subtitles built-in, but you can use the "Live Caption" feature of Google Chrome for very good real-time English captions.

You can return the course for a full refund within 14 days from your date of purchase. If you get started on the lessons and find out that they're not quite the right fit for you, just send me a message at [email protected] in that time window and I'll issue your refund.

After 14 days from your date of purchase, the course can no longer be returned.