These courses are aimed roughly 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.

The most important prerequisite is comfort with vectors and the basic tools of vector calculus. For example, you should know how to expand a vector in a basis, take dot products, and compute the gradient of a scalar function:

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

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

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

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 be familiar with calculus-based Newtonian mechanics, including $\vec F = m\vec a$. Some exposure to electric and magnetic fields will also come in handy, especially for later topics in the series.

For Part III, it will also help if you’ve seen some special relativity before, but again, you don’t have to be an expert.

And because this is a self-paced online course, you can always pause, review whatever background material you need, and then pick up right 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

Lesson 1: Beyond Cartesian Coordinates

• The need for general coordinate systems $x^i = (x^1, x^2, x^3)$. Coordinate systems defined. Cartesian vs curvilinear coordinates. Coordinate transformations. Coordinate ranges. Coordinate curves. Coordinate surfaces. Coordinate basis vectors $\vec e_i = \partial \vec r/\partial x^i.$ Orthonormal vs non-orthonormal basis vectors. Metric components $g_{ij} = \vec e_i \cdot \vec e_j.$ Displacement vector $\mathrm d\vec r = \mathrm dx^i \vec e_i.$ Line element $(\mathrm ds)^2 = g_{ij} \,\mathrm dx^i \mathrm dx^j.$ Polar, cylindrical, spherical coordinates.

Lesson 2: A Little Linear Algebra

• Non-orthonormal vector space basis. Dual basis $\vec e\,^i \cdot \vec e_j = \delta^i_j$. Generalized index notation. Upper vs lower indices. Basis expansion $\vec A = A^i \vec e_i$. Upper "contravariant" components $A^i = \vec A \cdot \vec e\,^i$. Dual basis expansion $\vec A = A_i \vec e\,^i$. Lower "covariant" components $A_i = \vec A \cdot \vec e_i.$ Dummy vs free indices. Index notation rules. Raising and lowering indices. Dot product $\vec A \cdot \vec B = A^i B_i = A_i B^i = g_{ij} A^i B^j = g^{ij} A_i B_j.$

Lesson 3: Vectors in Curvilinear Coordinates

• General coordinates $x^i = (x^1,x^2,x^3)$ as scalar fields. Trajectory $x^i(t)$. Velocity vector $\vec v = \frac{\mathrm{d} x^i }{\mathrm{d} t } \vec e_i.$ Dot notation. Subtleties of the position and acceleration vectors $\vec r$ and $\vec a$. Gradient vector $\vec \nabla f = \vec e\,^i \partial_i f.$ Gradient operator $\vec\nabla = \vec e\,^i \partial_i$. Dual coordinate basis vectors as normals to coordinate surfaces. $\vec e\,^i = \vec\nabla x^i$. Orthogonal coordinate systems. Scale factors. Position, velocity, and acceleration in polar coordinates.

Lesson 4: Tensors in Curvilinear Coordinates

• Review of tensors. Rank 2 tensor bases. Upper and lower components. Tensor dot product $T^{ij}S_{ij}.$ Raising and lowering indices. Tensor contractions. Metric tensor. Parallelepiped volume function. Anti-symmetric tensor $\epsilon_{ijk} = \sqrt{g}\, \varepsilon_{ijk}$. Wedge product. Cross product in a general basis $\vec A \times \vec B = \sqrt{g}\, \varepsilon_{ijk} A^i B^j \vec e\,^k.$ Volume element in general coordinates $\mathrm dv = \sqrt{g}\, \mathrm d x^1 \mathrm d x^2 \mathrm d x^3.$

Lesson 5: Tensor Calculus I

• Tensor fields. Directional derivative $\nabla_{\vec V} \boldsymbol T = \vec V \cdot \vec\nabla \boldsymbol T.$ Gradient tensor $\vec\nabla \boldsymbol T$. Differential $\mathrm d \boldsymbol T = \mathrm d \vec r \cdot \vec\nabla \boldsymbol T$. Gradient and directional derivative operators. Connection coefficients/Christoffel symbols $\Gamma_{ij}^k.$ Covariant derivative of a vector field $\nabla_i A^j = \partial_i A^j + \Gamma^j_{ik} \:\! A^k.$ Gradient and directional derivative of a vector field. Connection coefficient properties. Computing $\Gamma^k_{ij} = \frac{1}{2} g^{kl} (\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}).$ Why $\Gamma^k_{ij}$ is not a tensor. Connection coefficients for polar and spherical coordinates.

Lesson 6: Tensor Calculus II

• Directional derivative properties. Directional derivative and gradient in a general basis. Covariant derivatives of higher rank tensor fields $\nabla_i T^j{}_k = \partial_i T^j{}_k + \Gamma^j_{il} T^l{}_k - \Gamma_{ik}^l T^j{}_l.$ Covariant derivative of the metric tensor. Comma/semi-colon notation. Covariant derivative properties. Vector commutator $[\vec A, \vec B] = \nabla_{\vec A} \vec B - \nabla_{\vec B} \vec A.$ Non-commutativity of covariant derivatives $(\nabla_i \nabla_j - \nabla_j \nabla_i) A^k = R^k{}_{lij} A^l$. Riemann curvature tensor $R^k{}_{lij} = \partial_i \Gamma^k_{jl} - \partial_j \Gamma^k_{il} + \Gamma^k_{im}\Gamma^m_{jl} - \Gamma^k_{jm}\Gamma^m_{il}.$ Electric dipole moment $\vec p$. Electric force on a dipole $\vec F = \vec p \cdot \vec\nabla \vec E.$ Polarizability tensor $\vec p = \boldsymbol \alpha \cdot \vec E.$

Lesson 7: Tensor Tech Tools

• Computer tools for tensor calculations. Mathematica basics. Coordinates, basis vectors, metric in Mathematica. Diffgeo package. Connection coefficients. Anti-symmetric tensor. Raising/lowering indices. Contractions. Covariant derivatives. Contraction of connection coefficients $\Gamma^k_{ik} = \frac{1}{\sqrt{g}} \partial_i \sqrt{g}.$

Lesson 8: $\boldsymbol{\vec F = m \vec a}$ in Curvilinear Coordinates

• Review of $\vec F = m \vec a.$  Velocity in general coordinates $\vec v = \dot x^i \vec e_i.$ Acceleration in general coordinates $\vec a = (\ddot x^i + \Gamma^i_{jk} \dot x^j \dot x^k) \vec e_i.$ Generally covariant form of $\vec F = m \vec a.$ Coordinate vs orthonormal basis. Geodesic equation $\ddot x^i + \Gamma^i_{jk} \dot x^j \dot x^k = 0.$ Time derivatives of tensor fields. Parallel transport. $\vec F = m \vec a$ in spherical coordinates. Conservative force $\vec F = - \vec\nabla U.$ Equations of motion $m(\ddot x^i + \Gamma^i_{jk} \dot x^j \dot x^k) = - g^{ij} \partial_j U.$ Lagrangian formulation.

Lesson 9: Vector Operators in Curvilinear Coordinates

• Review of gradient, divergence, curl, Laplacian. Divergence in general coordinates $\vec\nabla \cdot \vec A = \frac{1}{\sqrt{g}} \partial_i (\sqrt{g} A^i).$ Laplacian in general coordinates $\nabla^2 f = \frac{1}{\sqrt{g}} \partial_i (\sqrt{g} \,g^{ij} \partial_j f).$ Curl in general coordinates $\vec\nabla \times \vec A = \frac{1}{\sqrt{g}} \vec e_i\, \varepsilon^{ijk} \partial_j A_k.$ Chain rule $\partial_{i'} f = (\partial_{i'} x^i) \partial_i f.$ Gradient, divergence, curl, and Laplacian in spherical coordinates. Electric and magnetic potentials, $\vec E = - \vec\nabla V - \partial_t \vec A,$ $\vec B = \vec\nabla \times \vec A.$ Gauge transformations. Lorenz gauge. Electromagnetic wave equations. Strain tensor $S_{ij} = \frac{1}{2} (\nabla_i \xi_j + \nabla_j \xi_i)$. Expansions and shears.

Lesson 10: General Coordinate Transformations

• Change of basis $\vec e_{i'} = \vec e_i (\mathsf M)^i{}_{i'}$. Vector transformation laws $A^{i'} = (\mathsf M^{-1})^{i'}{}_iA^i$ and $A_{i'} = A_i(\mathsf M)^i{}_{i'}.$ General coordinate transformations. Change of coordinate basis $\vec e_{i'} = \partial_{i'} x^i \vec e_i.$ Vector coordinate transformation laws $A^{i'} = \partial_i x^{i'} A^i$ and $A_{i'} = \partial_{i'}x^i A_i.$ Tensor transformation law $T^{i'j'} = \partial_ix^{i'}\partial_jx^{j'} T^{ij}.$ Transformation law "definition" of a tensor. Invariance of contractions. Jacobian $J = \det([\partial_{i'}x^i]).$ Tensor densities. Volume element from Jacobian. Connection coefficient transformation law.

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 $R.$

Parts I and II are already available in full, and you'll be able to begin those courses immediately after enrolling.

Part III is still under construction, and I’ll gradually be releasing those lessons starting later this year. By pre-ordering, you'll get access to each new lesson as soon as it's released.

I hope to have all the lessons for Part III completed in 2027, but I can't provide a definite date.

I think these courses 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 the lessons focus 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. I'll also introduce you to the Riemann curvature tensor, which is the mathematical representation of curvature that plays an essential role in general relativity.

Yes! A lot of them.

Solving problems is essential for learning just about anything involving physics or math, and tensor calculus is no exception.

The problems will give you an opportunity to practice with the tools we've developed in class, and they'll also introduce additional important topics and physics applications.

And you'll never be stuck! Every practice problem comes with a full, detailed solution, so you'll always have a way to check your work.

Part I: Fundamentals of Cartesian Tensors

• Lesson videos (14 hours)
• Practice problem + solution PDFs (308 pages)
• Lecture note PDFs (85 pages)
• Lecture slide PDFs (471 pages)
• Formula sheet PDF (6 pages)

Part II: Fundamentals of Curvilinear Tensors

• Lesson videos (12 hours)
• Practice problem + solution PDFs (241 pages)
• Lecture note PDFs (120 pages)
• Lecture slide PDFs (409 pages)
• Formula sheet PDF (7 pages)

Part III: Fundamentals of Spacetime Tensors

Coming soon! Planned materials include:

• Lesson videos
• Practice problem + solution PDFs
• Lecture note PDFs
• Lecture slide PDFs
• Formula sheet PDF 

At the pace of an intensive college class, each course would likely take at least 6 weeks to complete. But I encourage you to take them at a more comfortable pace of at least 10 weeks per course to give yourself more time to absorb the material—that's about one lesson and problem sheet per week.

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 timetable.

No! YouTube videos are great for quickly getting a taste of a topic and hopefully inspiring you 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 working through the material and solving lots of practice problems, and that’s what we’ll do in these courses.

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

That said, I can’t promise to provide detailed answers to every question, due to the number of students enrolled in the course.

For more personalized support, I offer an optional 1-to-1 Coaching add-on package, providing priority support to a limited number of students at a time. You’ll have the option to add that package after purchasing the course, if spots are currently available.

Yes—but I don’t generally recommend it for most students.

Each part of Fundamentals of Tensor Calculus builds on what came before. Part II assumes the foundation developed in Part I, and Part III assumes the tools developed in Parts I and II.

Even if you’ve already learned some of the prerequisite material elsewhere, each course in this series takes a unique approach to tensors that is likely different from what you’ve seen before. So if your goal is to build a complete understanding of tensor calculus, the complete series is the best path.

The complete series is also by far the better deal. Purchasing the three courses individually would cost $1,041, compared to just $694 for the complete series, including the bonuses.

That said, if you only want to enroll in one part for now, you can do so using the links below.

Note: Part III is still under construction, so it will not be available as a separate purchase until the full course is complete.

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

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.

Part III is still under construction—I plan to begin gradually rolling out those lessons later this year. By pre-ordering the course, you'll get access to each new lesson as soon as it's released.

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.

Yes. The checkout page includes a payment plan option of $269 per month for 3 months, for a total of $807.

The payment plan gives you access to the same course materials as the one-time payment option, but lets you split the purchase into three smaller monthly payments.

For the lowest total price, however, choose the one-time payment option at checkout—it's the best overall value.

You can return the courses for a full refund within 14 days from your date of purchase. Just send me a message at elliot@physicswithelliot.com within that time window, and I'll issue your refund. You'll need to delete any materials you may have downloaded from the course website. 

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