Lesson 1: Review Material

Outline of the Course:

Review Material covers basic calculus up to ordinary differential equations,  some linear algebra, and basic Newtonian mechanics.

Vector Calculus covers linear mappings and the mathematics of scalar and vector fields.

Newtonian Gravitational Fields covers the idea of the gravitational field from a vector calculus point of view and introduces the Poisson and Laplace equations.

The Language of Tensors explores the use of tensors and other geometrical objects in classical mechanics.

Maxwell’s Equations covers the electric field, the magnetic field, electromagnetic induction, the derivation of the Maxwell equations, and the electromagnetic tensor.

Applications of Maxwell’s Equations covers how to use the equations.

Basic Continuum Mechanics covers the ideas of deformation, strain, kinematics, stress, balance equations, constitutive equations, and symmetry groups.

Basic Thermodynamics covers thermodynamic systems, thermodynamic laws, internal energy, heat and work, thermodynamic potentials, entropy, the ideal gas, energy balance, stress-strain-temperature coupling, and reversible and irreversible processes.

Elastic Solids covers the equations of elasticity, equilibrium solutions, and elastic deformations.

Fluid Dynamics covers the perfect fluid, steady flow, Newtonian fluids, the Navier-Stokes equation, dimensional analysis, and viscous flow.

Waves in Deformable Media covers PDEs, ordinary waves, linearized waves, and shock waves.

Introduction to Special Relativity covers the speed of light, spacetime, and the objects that live there.

Introduction to Differential Geometry covers topology, manifolds, vectors, one-forms, and tensors from a mathematical point of view.

More about Special Relativity covering worldline geometry, velocity addition, acceleration, observers, the Frenet-Serret frame, the Levi-Civita tensor in spacetime, decomposing 2-forms, and differentiation in spacetime.

More About Special Relativity extends our understanding of topological groups, Lie groups and Lie algebras, the Lorentz group, one-parameter subgroups, and isometries.

Classical Mechanics covers configuration space, the action principle, Lagrangian mechanics, symmetries and conservation laws, the action principle in spacetime, and Hamiltonian mechanics

Dynamical Systems covers phase space, flows, autonomous systems, vector fields on phase space, fixed points, stability, linearization, Lyapunov functions and Lyapunov stability, periodic orbits, limit cycles, the Poincaré-Bendixson theorem, bifurcations, conservative vs dissipative systems, sensitivity to initial conditions, chaos, connections to physics.

Fields in Terms of Special Relativity covers a general theory of fields in the language of special relativity in terms of Lagrangian mechanics, action in fields, relativistic fields, the relativistic Lagrangian, the scalar potential, the vector potential, and the electromagnetic field tensor.

Particles and Fields covers how fields interact with particles, how particles interact with fields, the equations of motion in a field, gauge invariance, and the Lorentz force.

Still More About Special Relativity explores physics in flat spacetime including the stress-energy tensor, four-momentum conservation, and matter in spacetime.

Electromagnetic Radiation and Optics covers geometric optics as a mechanics problem, Liénard-Wiechert potentials, radiation energy, dipole radiation, plane waves, reflection and refraction, superposition, wave packets, and waves in conductors.

Advanced Ideas of Special Relativity covers more physics in flat spacetime.

Classical Statistical Mechanics covers the desire for a microscopic description, ensembles, Liouville’s theorem, entropy, the postulate of a priori probabilities, temperature form the microcanonical ensemble, the Boltzmann factor, the partition function and the thermodynamic potentials, and ergodicity.

Kinetic Theory covers ideal gases, molecular chaos, the distribution function, the Liouville equation and its reduction to the Boltzmann equation, the Maxwell-Boltzmann relation, pressure due to molecular collisions, mean free path, viscosity, thermal conductivity, diffusion, H-theorem, connection to thermodynamics and hydrodynamics, and the limitations.

The Concept of Plasmas covers the particle kinetics approach to plasmas.

Magnetohydrodynamics covers the theory of plasmas as a fluid.

Astrophysics includes stellar structure, stellar evolution, interstellar medium, and galaxies.

More Differential Geometry covers curved manifolds, tensor algebra in curved manifolds, commutators, one-parameter subgroups of diffeomorphisms, geodesics, parallel transport, covariant derivatives, and the geodesic equation.

More Differential Geometry covers geodesic deviation, the Riemann curvature tensor, and Bianchi identities.

More Differential Geometry covers local frames, proper frames, curvature, and differential forms.

Physics in Curved Manifolds covers Poisson brackets, geodesics, curvature, thermodynamics, fluid dynamics, and electromagnetic fields.

Introduction to General Relativity, covering the equivalence principle, fluid dynamics in a gravitational field, properties of covariant derivatives, factor ordering, the precession of equinoxes, the pendulum, Deriving the field equations—path 1.

General Relativity, including the Newtonian limit, the linearized theory, the Schwarzschild solution, and post Newtonian theory.

More General Relativity, including trajectories near a massive spherical body, Killing vectors and conservation laws, and asymptotic flatness.

The Variational Approach introduces the Einstein-Hilbert action, general variance, more about the stress-energy tensor, the second path to the field equations, covariance and invariance, the tetrad, and the initial-value problem.

Other approaches covers the Palatini formalism, the route from Lovelock’s theorem, thermodynamic gravity, and the gauge theory of gravity.

Quantum Mechanics covers the quantum of action, wave-particle duality, the de Broglie relation, the Schrödinger equation, the wave function, the probability interpretation and the Born rule, operators and observables, commutation relations and uncertainty, stationary states, the classical limit, simple solvable systems, phase space and Hilbert space, Poisson brackets, entanglement, decoherence, and the measurement problem.

Statistical Mechanics covers the density operator and quantum ensembles, the quantum microcanonical ensemble, the density matrix for thermal states, the partition function, Bose-Einstein statistics, Fermi-Dirac statistics, Bose gases, Bose-Einstein condensation, Fermi gas, degeneracy pressure, blackbody radiation, the classical limit, lasers, and superconductivity.

Quantum Field Theory covers the failure of quantum mechanics at relativity, scalar fields and the Klein-Gordon equation, canonical quantization of fields, , creation and annihilation operators, particles as excitations, the vacuum state and zero-point energy, the Feynman propagator and causality, simple interacting theories, gauge fields and the photon, fermions and the Dirac field, Fock space, the Casimir effect, spontaneous emission, particle creation, renormalization, infinities, and the measurement problem in QFT.

The Standard Model of Particle Physics covers quarks, leptons, generations, color, weak isospin, gauge bosons such as photons, W and Z, gluons, the Higgs boson, U(1) EM, SU(2) × U(1) Electroweak theory, QCD, the Higgs mechanism, Symmetry breaking, masses and mixing, Feynman rules, the hierarchy problem, neutrino masses, dark matter, baryogenesis, and grand unification.

Nuclear Physics covers nuclear properties, nuclear forces and the strong interaction, the liquid drop model, the shell model, the collective model, radioactivity and decay processes, nuclear reactions, fission, fusion, nucleosynthesis, r-processes, s-processes, nuclear matter, exotic nuclei, the nuclear equation of state, and quark0gluon plasmas.

Star death covers the endpoints of stellar evolution, supernovae, white dwarf stars, and neutron stars.

Black Holes covers what is a black hole? what is gravitational collapse, black hole astrophysics, Schwarzschild black holes, Reissner-Nordstrom black holes, Kerr black holes, Kerr-Newman black holes, the physics of black holes.

Black holes physics includes singularities, horizons, the ergosphere, energy extraction, and Hawking radiation.

High energy astrophysics covers relativistic particles, compact objects as engines, accretion disks, supernova remnants, gamma-ray bursts, AGNs and quasars, cosmic rays, synchrotron radiation, inverse Compton scattering, bremsstrahlung, pair production and annihilation, and neutrino astrophysics.

Cosmology covers the cosmological principle, ,cosmological red shift, the evolution of the universe, and cosmic horizons, Friedmann universes, the cosmological constant, the hot big bang model, de Sitter space, blackbody radiation, CMBR, ΛCDM model, and recent results from JWST.

The Early Universe covers radiation and temperature, the scale factor, the radiation era, the isotropic CMB and the horizon, and anisotropies of the CMBR.

History of the Universe covers condensation into galaxies, into stars, into atoms, into nuclei, and into nucleons.

Frontiers of Cosmology includes inflation, inflation via scalar fields, structure formation, dark matter, and dark energy.

Gravitational Waves include gravitational plane waves, sources of gravitational waves, LIGO and other detectors.

Introduction

The path to advanced theoretical physics is not built on forgetting the foundations, but on mastering them so thoroughly that they become part of your intuition. You are about to embark on a journey into Classical Field Theory and Relativity. This course will ask you to think deeply, calculate carefully, and develop real physical intuition. Before we construct the beautiful structures of tensors, fields, and curved spacetime, we must ensure the foundation is strong and reliable.

The purpose of this lesson is to sharpen the essential mathematical and physical tools you will use throughout the entire course. Treat this not as mere revision, but as active preparation of your working toolkit. Every topic here has been chosen because it will reappear—often in more powerful and surprising forms.

Differential Calculus

Differential calculus is the mathematics of instantaneous change—nature’s way of revealing how one quantity responds to an infinitesimal shift in another.

Intervals of Functions

An interval is a connected subset of the real line. The main types are open (a,b), closed [a,b], half-open, and unbounded. Physical quantities live on specific intervals—the choice of interval affects continuity, differentiability, and physical meaning.

Limits of Functions and Continuity

The concept of a limit is one of the most important and subtle ideas in all of mathematics. It formalizes the intuitive notion of “getting closer and closer” to a particular value without necessarily reaching it. In physics, limits appear everywhere: the instantaneous velocity of a particle as the time interval shrinks to zero, the electric field very close to a point charge, the behavior of a wave as distance or time goes to infinity, and the approach to a singularity in spacetime.

Definition 1.1:

(1.1)

We say that if, as x approaches a (from within the allowed interval), the value of f(x) becomes arbitrarily close to L, no matter how close we require it to be.

Definition 1.2 One-Sided Limits:

The right-hand limit considers x approaching a from values greater than a.  

The left-hand limit considers x approaching a from values less than a.

For the ordinary (two-sided) limit to exist, both one-sided limits must exist and be equal.

Definition 1.3 ε-δ Definition:

The limit equals L if, for every ε>0, there exists a δ>0 such that

(1.2)

Definition 1.4 Continuity at a Point:

A function f is continuous at x=a if three conditions hold:

f(a) is defined,

exists,

.

Definition 1.5 Continuity on an Interval:

A function is continuous on an interval if it is continuous at every point in that interval.

Principle 1.1: Limits allow us to study the local behavior of a function rigorously using information from its immediate neighborhood. Almost all of differential calculus and field theory rests on this idea.

Principle 1.2: Continuity means there are no sudden jumps or breaks in the function. Most quantities we encounter in classical physics (position, velocity, density, potential, temperature) are continuous functions of space and time.

Principle 1.3: Discontinuities are not always defects—they often signal important physical phenomena such as shock waves, phase transitions, or idealized boundaries.

Principle 1.4: When working on a physical problem, always ask yourself: “Is this quantity continuous here? What happens as we approach a boundary or a special point?”

The Derivative

(1.3)

It gives instantaneous rate of change and the best linear approximation to the function.

Differentiation Rules

The Constant Rule

(1.4)

The Constant Multiple Rule

(1.5)

The Sum/Difference Rule (Linearity)

(1.6)

The Power Rule

(1.7)

The Reciprocal Rule

(1.8)

Root Rule

(1.9)

Product Rule

(1.10)

Quotient Rule

(1.11)

Chain Rule

(1.12)

The Chain Rule is perhaps the most important and frequently used differentiation rule in all of theoretical physics. It appears whenever one quantity depends on another that itself depends on a third (position depends on time, potential depends on position, field depends on coordinates, etc.). Mastering the Chain Rule is essential for Lagrangian and Hamiltonian mechanics, field theory, and relativity.

Calculating Derivatives

Polynomial Functions

Polynomials are the simplest and yet most useful functions in theoretical physics—they serve as local approximations to almost every smooth physical quantity. A polynomial function is any function of the form

(1.13)

where n is a non-negative integer and the are constants (with ).

Because differentiation is a linear operation, you may differentiate term by term. Combine this with the Power Rule,

(1.14)

General Procedure:

Write the polynomial in standard form (highest power first).

Differentiate each term separately using the Power Rule.

Apply the Constant Rule to any constant term (its derivative is zero).

Simplify the resulting expression.

Rational Functions

Rational functions appear naturally whenever one quantity is divided by another—potentials, densities, velocities, and many field expressions are ratios of polynomials. A rational function is a ratio of two polynomials:

(1.15)

where p(x) and q(x) are polynomials and q(x)≠0.

To differentiate a rational function, we apply the quotient rule,

(1.16)

Always simplify the expression before and after differentiating when possible. Factoring, canceling common terms, and writing the function in a simpler form often makes the derivative much cleaner and easier to interpret physically.

General Procedure:

Simplify the rational function as much as possible (cancel common factors).

Apply the Quotient Rule.

Simplify the resulting numerator if possible.

Write the final answer in factored or reduced form.

Trigonometric Functions

Trigonometric functions describe periodic and oscillatory phenomena—from waves and rotations to the geometry of spacetime itself. The standard trigonometric functions and their derivatives are essential tools you will use throughout this course, especially when working with waves, oscillations, circular motion, and angular quantities in relativity.

(1.17)

(1.18)

(1.19)

(1.20)

(1.22)

(1.23)

These derivatives are most useful when combined with the Chain Rule.

Exponential Functions

(1.24)

Logarithmic Functions

(1.25)

Hyperbolic Functions

Hyperbolic functions are the natural language of special relativity—they replace ordinary trigonometry when we work with rapidities instead of angles. The hyperbolic functions (sinh, cosh, tanh, etc.) are defined in terms of exponentials and serve as the relativistic analogues of the ordinary trigonometric functions. They appear naturally whenever we deal with Lorentz boosts, proper time, and Minkowski spacetime geometry.

(1.25)

(1.26)

(1.27)

(1.28)

(1.29)

(1.30)

Implicit Differentiation

Many of the most important relations in physics are not given as explicit functions, but as equations that define one variable in terms of others. Implicit differentiation is the tool that lets us find rates of change without solving for the variable explicitly. Very often in theoretical physics a quantity is defined by an equation of the form F(x,y)=0 rather than by an explicit formula y=f(x). In such cases we use implicit differentiation.

General Procedure:

Differentiate both sides of the equation with respect to x, treating y as a function of x (so you must use the Chain Rule on any term containing y).

Collect all terms having dy/dx.

Solve for dy/dx.

If F(x,y)=0, then,

(1.31)

Implicit differentiation is indispensable because many fundamental physical laws (constraints, conservation laws, field equations, and surfaces of constant energy or potential) are given as implicit relations rather than explicit functions. The technique allows you to extract rates of change and slopes directly from the governing equation without solving for one variable in terms of the others — a skill that becomes extremely powerful when you reach Lagrangian mechanics, general relativity, and quantum field theory.

Topics in Differentiation

The Mean Value Theorem for Derivatives

Between any two points on a smooth curve, there is at least one point where the instantaneous rate of change exactly equals the average rate of change over the whole interval.

Theorem 1.1 Mean Value Theorem for Derivatives:
Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Then there exists at least one point c in (a, b) such that

(1.32)

In other words, there is a point where the instantaneous rate of change (the derivative) exactly equals the average rate of change over the entire interval.

The Mean Value Theorem is the fundamental bridge between local information (the derivative at a point) and global information (the net change over an interval). It is one of the most important theorems in all of calculus because it connects the microscopic behavior of a function to its macroscopic behavior. Nearly every major result in analysis, differential equations, and theoretical physics ultimately rests on this bridge.

The Linear Approximation

(1.33)

Newton’s Method

When we need to solve an equation that cannot be solved algebraically, Newton’s Method gives us a powerful way to obtain increasingly accurate numerical solutions by using the derivative as a guide. Start with an initial guess (preferably close to the expected root). Then generate the next approximation by the rule

(1.34)

Repeat until the change is smaller than a desired tolerance. Under good conditions, the method converges quadratically—the number of correct digits roughly doubles with each iteration.

Logarithmic Differentiation

When a function is a complicated product, quotient, or power, taking the logarithm often turns multiplication into addition and exponents into multiplication—dramatically simplifying the differentiation. Logarithmic Differentiation is a powerful technique used when direct differentiation of a function would be messy or difficult. It is especially useful for products of many functions, quotients with variable exponents, and functions of the form [f(x)]g(x).

General Procedure:

Take the natural logarithm of both sides:

(1.35)

Differentiate both sides with respect to x (using the Chain Rule on the left)

(1.36)

Solve for dy/dx.

(1.37)

Second- and Higher-Order Derivatives

The first derivative tells us how a quantity changes; the second derivative tells us how that change itself changes. Higher derivatives reveal ever finer details of a function’s behavior.

The second derivative of f(x) is the derivative of the first derivative.

The third derivative of f(x) is the derivative of the second derivative.

And so on.

Principle 1.5: The second derivative measures how rapidly the first derivative is changing. It quantifies concavity (upward or downward curvature) of a function.  

Principle 1.6: Higher-order derivatives become essential in Taylor expansions, which allow us to approximate complicated functions locally by polynomials. This technique underlies perturbation methods used throughout field theory and relativity.

Principle 1.7: The existence and behavior of higher derivatives tell us how smooth a physical quantity is. Discontinuities in higher derivatives often signal important physical transitions (e.g., shock waves).

Stationary Points

The points where a function stops changing—even momentarily—are often the most physically significant locations in a system. A point x=c is a stationary point (also called a critical point) of a differentiable function f(x) if

(1.38)

At a stationary point the tangent line is horizontal—the instantaneous rate of change is exactly zero.

Classification of Stationary Points:

First Derivative Test: Examine the sign of f'(x) on either side of c. If it changes from positive to negative it is a local maximum. If it changes from negative to positive it is a local minimum. No change indicates that there is no min or max, it may be an inflection point or a flat region, but there is no way top know from this test.

Second Derivative Test: If you have a stationary point examine f''(c). If f''(c)>0 then we have a local maximum. If f''(c)<0 then we have a local minimum. If f''(c)=0 then the test is inconclusive, use the first derivative test.

Curvature

(1.39)

a first taste of geometric ideas.

Integral Calculus

Integration is nature’s way of accumulating infinitesimal contributions into a finite, meaningful result. Integration reverses differentiation and computes accumulated quantities.

Riemann Sums

The definite integral is defined as the limit of a sum. Riemann sums provide the concrete way to build that sum by approximating the area under a curve using rectangles (or other simple shapes).

Divide the interval [a, b] into n subintervals, each of width

(1.40)

In each subinterval , choose a sample point and form a rectangle with height and width Δ x. The area of the ith rectangle is .

Definition 1.6 Riemann Sum: The Riemann sum for the function f(x) on [a, b] is

(1.41)

Different choices of give different Riemann sums:

Left Riemann sum: (left endpoint).

Right Riemann sum: (right endpoint).

Midpoint Riemann sum: is the midpoint of the subinterval.

Upper and lower sums: using the maximum and minimum values in each subinterval.

As n increases and Δ x becomes smaller, the rectangles fit the curve more closely. The Riemann sum becomes a better and better approximation to the true area under the curve.

Principle 1.8: A Riemann sum turns a continuous accumulation problem into a finite sum—something we can actually compute.

Principle 1.9: The accuracy of the approximation improves as the width of the rectangles Δ x approaches zero.

Principle 1.10: The definite integral is defined as the limit of the Riemann sum as n→∞ and Δ x→0
(provided the limit exists).

Principle 1.11: Different choices of sample points (left, right, midpoint) may give slightly different approximations, but they all converge to the same value for continuous functions.

The Definite Integral

The definite integral is the precise mathematical embodiment of accumulation — the limit of summing infinitely many infinitesimal contributions.

Definition 1.7 Definite Integral: The definite integral of a function f(x) from a to b is defined as

(1.42)

provided this limit exists and is independent of how we choose the sample points .

Principle 1.12: The definite integral is the natural generalization of summation to the continuous case. It turns “adding up many small pieces” into a precise limiting process.

Principle 1.13: The value of the integral depends only on the function and the endpoints—not on the particular choice of Riemann sum (left, right, midpoint, etc.), as long as f is well-behaved (e.g., continuous).

Principle 1.14: The definite integral is a global object: it accumulates information over an entire interval, in contrast to the derivative, which is a local object.

Principle 1.15: Almost every major law in theoretical physics can be written either as a differential equation (using derivatives) or as an integral equation (using definite integrals). Mastery of both is essential.

Antiderivatives

If the derivative tells us how a quantity changes, the antiderivative answers the reverse question: what quantity could have produced this rate of change? An antiderivative of a function f(x) is a function F(x) such that

(1.43)

Adding an arbitrary constant transforms the antiderivative into an indefinite integral.

The Fundamental Theorem of Calculus

Connects derivatives, definite integrals, and antiderivatives

(1.44)

where F'(x)=f(x).

Integration by Substitution

Substitution is the reverse of the Chain Rule—it lets us simplify a complicated integral by changing to a new variable that matches the structure of the integrand.

General Procedure:

Look for a part of the integrand that is the derivative of another part.

Choose u to be the “inside” function (the one being differentiated).

Compute du.

Rewrite the entire integral in terms of u.

Integrate with respect to u.

Substitute back into x.

Add the constant of integration.

Principle 1.16: Substitution reverses the Chain Rule. Wherever you see a composite function whose derivative appears (even as a factor), substitution is likely the right tool.

Principle 1.17: The success of substitution depends on recognizing the inner function and its derivative in the integrand. Training your eye to spot this pattern is one of the most practical skills in calculus.

Principle 1.18: Many integrals in theoretical physics (action integrals, potentials, relativistic kinematics, probability densities) become tractable only after a clever substitution.

Principle 1.19: When a substitution works, the integral usually simplifies dramatically. If it becomes more complicated, try a different choice of u.

Integration by Parts

Integration by parts is the reverse of the Product Rule—it allows us to trade a difficult integral for one that is hopefully simpler.

(1.45)

General Procedure:

Choose u and dv such that u is a function the becomes simpler when differentiated, and dv is easy to integrate.

Compute du and v.

Apply (1.45).

Rewrite the entire integral in terms of u.

If the new integral is easier to calculate, proceed, otherwise try swapping the choice of u and dv.

A helpful rule to choose your function is called the LIATE Rule. The easiest is Logarithmic, then Inverse Trig, then Algebraic Polynomials, then Trig, then Exponential.

Principle 1.20: Integration by parts reverses the Product Rule, just as substitution reverses the Chain Rule.

Principle 1.21: The success of the method depends on a wise choice of u and dv. The goal is to make the new integral ∫v du simpler than the original.

Principle 1.22: This technique is indispensable in theoretical physics—it appears in deriving energy integrals, solving differential equations, and the like.

Principle 1.23: When used skillfully, integration by parts can transform an impossible integral into one that is elementary or even zero.

Basic Calculus in Mathematica

Back-of-the-Envelope Calculations

Order-of-magnitude reasoning

The ability to estimate orders of magnitude is one of the most valuable skills a theoretical physicist can possess. It lets you quickly decide whether an idea is plausible long before performing an exact calculation. Order-of-magnitude reasoning (also called back-of-the-envelope calculation) is the art of finding an approximate answer that is correct to within a factor of 10 or so—accurate enough to reveal whether a physical idea makes sense. We deliberately sacrifice precision for speed and insight.

Fermi Problems

To estimate the number of piano tuners in New York City.

Look up the population of New York ≈ people.

Fraction of households with pianos ≈ 1/100 → pianos.

Each piano tuned once per year → tunings per year.

One tuner can service ≈ 1000 pianos per year → roughly 100 piano tuners.

Dimensional Analysis

Use the dimensions (abstract units) of physical quantities to discover relationships and check consistency.

If you know a quantity depends on certain variables, you can often guess the functional form up to a dimensionless constant.

For example, the period of a simple pendulum has abstract units of time.
The period T can only depend on length ℓ, gravity g, and mass m. The only combination with dimensions of time is

(1.46)

since the exact result is , we are off by a factor of 2 π.

Principle 1.25: An order-of-magnitude estimate is successful if it is within a factor of 10 of the true value. This level of accuracy is usually enough to decide whether a physical mechanism is important or negligible.

Principle 1.26: Order-of-magnitude reasoning exposes mistakes instantly. If your estimate differs from a known value by many orders of magnitude, you have likely made a conceptual error.

Principle 1.27: Many breakthroughs in theoretical physics began with a simple back-of-the-envelope calculation (Fermi’s estimate of the Trinity test yield, Dirac’s estimate of the fine-structure constant, etc.).

Principle 1.28: Always combine order-of-magnitude estimates with dimensional analysis. Together they form a powerful consistency check before investing time in exact calculations.

Use Mathematica in this way:

Matrix Algebra

Up to this point you have worked primarily with single numbers. In theoretical physics we frequently need to handle systems that transform many quantities at once in a linear way. Matrices are the perfect tool for this task. They provide a compact, systematic notation for linear operations and appear everywhere from classical mechanics to quantum mechanics to relativity and differential geometry.

A matrix is a rectangular array of numbers (or symbols) arranged in rows and columns. An m×n matrix has m rows and n columns. We usually denote matrices by bold capital letters, A, B, etc.

Definition 1.8 Matrices: A matrix A with elements with  (row i, column j) is written as

(1.47)

When m=n we call the matrix square.

An m×1 matrix is called a column matrix.

An 1×n matrix is called a row matrix.

A matrix with nothing by 0 entries is called the zero matrix.

A matrix with 1 on each diagonal element, and zeros everywhere else is called an identity matrix.

The transpose of a matrix A, denoted , is the matrix obtained by interchanging its rows and columns—the element that was in row i, column j of A moves to row j, column i of .

Principle 1.29: Matrices are a bookkeeping device that lets us handle many linear relationships simultaneously.  

Principle 1.30: The power of matrix algebra comes from its ability to generalize ordinary arithmetic to higher dimensions while preserving linearity.

Principle 1.31: Almost every linear physical law can be written in matrix form. Learning to read, manipulate, and interpret matrices is therefore essential for theoretical physics.

Principle 1.32: A matrix by itself is just a table of numbers; its true meaning appears when you apply it to a physical state and see how that state is transformed.

Matrix Arithmetic

Matrix arithmetic is straightforward once you understand that most operations are performed element by element, with one important exception (matrix multiplication, covered in the next subsection).

Let A and B be two matrices of the same size (same number of rows and columns).

(1.48)

(1.49)

(1.50)

Addition is commutative: A+B=B+A.

Addition is associative: (A+B)+C=A+(B+C).

Scalar multiplication distributes over addition: c(A+B)=c A+c B.

The zero matrix (all entries zero) acts as the additive identity: A+0=A.

Principle 1.33: Matrix addition and scalar multiplication obey the same algebraic rules as ordinary numbers (they form a vector space). This is why matrices are so useful in physics.

Principle 1.34: These operations are element-wise. They are simple but extremely powerful when applied to large systems (e.g., finite-element simulations, discretizations of field equations, etc .)

Principle 1.35: Addition of matrices corresponds to superposition of linear effects.

Principle 1.36: Always check that two matrices have the same dimensions before attempting to add or subtract them. Dimension mismatch is one of the most common (and easily avoided) sources of error.

You should be able to add, subtract, and scale matrices confidently by hand for small matrices (2×2 and 3×3) and in Mathematica for larger ones. This basic arithmetic is the foundation for everything that follows.

Matrix Multiplication

Matrix multiplication is fundamentally different from ordinary multiplication. It is not performed element by element. Instead, it encodes the composition of linear transformations.

Definition 1.9 Matrix Multiplication: Let A be an m×p matrix and B be a p×n matrix. The product C = A B is an m×n matrix whose elements are

(1.51)

Matrix multiplication is only defined when the number of columns of the first matrix equals the number of rows of the second matrix (the “inner dimensions” must match).

Matrix multiplication is associative: (A B)C=A(B C).

It is distributive over addition: A(B+C)=A B+A C.

It is not commutative: In general, A B≠B A.

The identity matrix I satisfies A I=I A=A.

Principle 1.37: Matrix multiplication represents the composition of linear transformations. Applying A followed by B is the same as applying the single transformation B A.

Principle 1.38: The non-commutativity of matrix multiplication (A B≠B A) is not a bug—it is a feature. It appears in rotations, Lorentz boosts, and quantum operators.

Principle 1.39: Many fundamental physical equations are written compactly using matrix multiplication: equations of motion in linear systems, coordinate transformations, change of basis, and the action of operators on state vectors.

Principle 1.40: Always check dimensions before multiplying. A dimension mismatch is usually a sign that your physical setup or coordinate choice needs rethinking.

Determinants

The determinant is a scalar that encodes profound geometric and physical information: it tells us how volumes transform under a linear mapping, whether a system of equations has a unique solution, and whether a transformation preserves orientation. The determinant of a square matrix A, written det A, or ∣A∣, is a single number that carries essential information about the linear transformation represented by the matrix.

Definition 1.10 Determinant: For a 2 × 2 matrix,

(1.52)

the determinant is

(1.53)

The absolute value ∣det⁡A∣ is the factor by which areas (in 2D) or volumes (in 3D) are scaled under the linear transformation defined by A. If ∣det⁡A∣=1, the transformation preserves volume. If det⁡A=0, the transformation collapses the space onto a lower-dimensional subspace (singular matrix).

Minors and cofactors are the building blocks that allow us to compute determinants systematically and to understand how a matrix behaves when we remove one row and one column — an operation that has deep physical meaning in systems with constraints.

Definition 1.11 Minor: Let A be an n×n square matrix. The minor of the element is the determinant of the (n−1)×(n−1) submatrix obtained by deleting row i and column j from A. In other words, is the determinant of the matrix that remains after removing the ith row and jth column.

Definition 1.12 Cofactor: The cofactor of element is the signed minor

(1.54)

The sign alternates in a checkerboard pattern starting with positive for .

They provide a systematic recursive way to compute determinants of large matrices (cofactor expansion). In physics, removing a row and column often corresponds to imposing a constraint or fixing a coordinate—minors naturally appear in such situations.

Principle 1.41: A minor tells you the “volume scaling factor” of the linear transformation restricted to the subspace obtained by removing one dimension.

Principle 1.42: The alternating sign in the cofactor accounts for orientation (whether the removal of that row and column preserves or reverses the handedness of the coordinate system).

Principle 1.43: Minors and cofactors are the foundation for Cramer’s Rule, matrix inversion, and the theory of linear dependence.

Principle 1.44: Although we rarely compute large determinants by hand (use Mathematica!), understanding what minors and cofactors represent gives deeper insight into the structure of linear systems and tensor operations.

In Mathematica we write:

Matrix Inversion

The inverse of a matrix is the mathematical expression of undoing a linear transformation—it tells us how to recover the original vector after the transformation has been applied. If a square matrix A represents a linear transformation, its inverse (when it exists) is the transformation that reverses the effect of A. Applying A followed by returns the original vector

(1.55)

Definition 1.13 Invertible Matrices: A square matrix A is invertible (or nonsingular) if there exists a matrix such that equation (1.55) holds. If no such matrix exists, A is singular.

A square matrix is invertible if and only if its determinant is nonzero.

The formula for the inverse uses the adjugate matrix (the transpose of the cofactor matrix).

(1.56)

Principle 1.45: A matrix is invertible precisely when its determinant is nonzero. A vanishing determinant means the transformation collapses information (linear dependence) and cannot be undone.

Principle 1.46: Matrix inversion corresponds to solving systems of linear equations. The equation A x=b has a unique solution if and only if A is invertible.

Principle 1.47: In physics, invertibility is deeply connected to the independence of coordinates or basis vectors. Singular matrices often signal constraints, degeneracies, or critical physical situations (e.g., resonance, loss of degrees of freedom).

Principle 1.48: For large matrices, compute the inverse using numerical methods or symbolic software rather than by hand. However, understanding what the inverse means physically is far more important than the mechanical calculation.

In Wolfram Language we write these operations.

Never invert a matrix unless you are certain it is invertible (det⁡ A≠0). In numerical work, small determinants can lead to large rounding errors—a common source of instability in simulations.

You should be able to compute inverses of 2 × 2 matrices by hand, understand when a matrix is invertible, and use Mathematica confidently for larger cases. Matrix inversion is a fundamental operation you will use repeatedly.

Vector Algebra

Arrow Geometry

Visualize vectors as directed line segments.

Resolving Vectors: Express a vector in terms of basis vectors

(1.57)

Vector Arithmetic

Vectors are the natural objects for representing quantities that possess both magnitude and direction. Learning to add, subtract, and scale them is the foundation for describing motion, forces, fields, and spacetime geometry. Vector arithmetic extends ordinary number arithmetic to quantities that have direction as well as magnitude. The operations are both geometrically intuitive and algebraically precise.

To add two vectors and , place the tail of at the head of . The sum is the vector from the tail of to the head of (called the parallelogram law). In terms of components we write

(1.58)

Vector subtraction is done this way

(1.59)

where is the vector with the same magnitude as but in the opposite direction.

Scalar multiplication is,

(1.60)

Vector addition is commutative: .

Vector addition is associative: .

Scalar multiplication distributes over vector addition.

The zero vector satisfies .

Principle 1.49: Vector addition corresponds to the physical principle of superposition, the combined effect of multiple vector quantities is their vector sum.

Principle 1.50: Scalar multiplication scales both the magnitude and (when negative) the direction of a vector—essential for describing opposite forces, negative velocities, or scaling fields.

Principle 1.51: Vector arithmetic is component-wise in a given basis. Changing the basis changes the components but not the physical vector itself.

Principle 1.52: Mastering vector arithmetic gives you the ability to move freely between geometric intuition (arrows) and algebraic calculation (components)—a skill you will use constantly in mechanics, electromagnetism, and relativity.

Scalar Product (Dot Product)

One of three ways to multiply vectors. The scalar product measures how much one vector projects onto another—it quantifies alignment, work, and the geometric relationship between directions in space. The scalar product, also called the dot product, of two vectors and is a single real number that encodes essential geometric and physical information.

(1.61)

where θ is the angle between the vectors.

(1.63)

Commutative: .

Distributive over addition.

.

If and neither vector is zero, then and are orthogonal (perpendicular).

Principle 1.53: The scalar product extracts the component of one vector that lies parallel to another. It measures “how aligned” two vectors are.

Principle 1.54: Orthogonality is one of the most important concepts in physics—it appears in normal modes, perpendicular forces, gauge conditions, and basis vectors.

Principle 1.55: The scalar product is the first example of an inner product. It generalizes naturally to the metric tensor in relativity and to abstract Hilbert spaces in quantum mechanics.

Principle 1.56: Always choose the most convenient form, use the geometric definition when you know magnitudes and angles; use components when working in coordinates.

Vector Product (Cross Product)

This is the second way to  multiply vectors. The vector product produces a new vector that is perpendicular to both original vectors—it encodes the idea of rotation, area, and the right-hand rule that appears throughout physics.

(1.63)

where θ is the smaller angle between the vectors (0≤θ≤π), and is the unit vector perpendicular to the plane of the two vectors, with direction given by the right-hand rule.

(1.64)

The cross product is anti-commutative.

It is distributive over addition.

.

Principle 1.57: The vector product measures the part of two vectors that is perpendicular—it quantifies “twist” or rotational character.

Principle 1.58: Unlike the scalar product, the vector product yields a vector, not a scalar, and its direction follows the right-hand rule (a convention deeply embedded in physics).

Principle 1.59: The magnitude gives the area of the parallelogram; this makes the vector product essential for defining oriented areas, torques, and angular momentum.

Principle 1.60: The vector product is a three-dimensional operation. In higher dimensions it generalizes to the exterior product (wedge product), which you will meet when working with differential forms in relativity and field theory.

Kinematics

Kinematics is the geometry of motion — it describes how things move without yet asking why. Kinematics is the branch of mechanics that deals with the description of motion—positions, velocities, accelerations, and trajectories—independent of the forces that cause the motion. It forms the foundation for all of dynamics and appears throughout classical mechanics, special relativity, and field theory.

Vector Functions of a Single Variable

A particle moving in space has a position vector that depends on time t. This is a vector-valued function

(1.65)

The first time derivative is velocity

(1.66)

The definite integral of the velocity gives the displacement

(1.67)

The time derivative of velocity is acceleration

(1.68)

Space Curves

The path traced by is a space curve. At any point we can define:

Unit tangent vector: .

Tangent line: The line through in the direction of .

Principal normal and curvature (reviewed in more depth in Topics in Differentiation and later in differential geometry in latter lessons).

Principle 1.61: Kinematics is purely descriptive: it answers “where?” and “how fast?” but not “why?”

Principle 1.62: All vector quantities in kinematics (position, velocity, acceleration) transform properly under changes of coordinate systems—a property that becomes critical in special and general relativity.

Principle 1.63: The derivative of position is velocity; the derivative of velocity is acceleration. This chain continues and forms the basis of Newton’s laws and Lagrangian mechanics.

Principle 1.64: Curvature of a space curve quantifies how sharply the path bends—a concept that generalizes directly to the curvature of spacetime in general relativity.

Partial Differentiation

Most quantities in the physical world depend on several independent variables simultaneously. Partial derivatives let us study how a function changes with respect to one variable while holding all the others fixed. In elementary calculus you studied functions of a single variable. In theoretical physics we almost always deal with functions of several variables—position depends on (x, y, z); temperature depends on position and time; potential energy depends on coordinates and possibly velocity. Partial derivatives are the natural tool for such multivariable situations.

Definition 1.14 Partial Derivative: Let f(x,y,z,… ) be a function of several variables. The partial derivative of f with respect to ( x ) (treating the other variables as constant) is

(1.69)

provided the limit exists. We use the symbol ∂ (a curly d) to remind ourselves that other variables are held fixed.

Ordinary Differential Equations

In the real world, many quantities change continuously over time or space according to rules that involve their own rates of change. The mathematical statement of such a rule is called a differential equation. Learning to work with differential equations is one of the most important steps in your journey toward theoretical physics. Almost every law of physics you will meet—from Newton’s second law to Schrödinger’s equation—is a differential equation.

Definition 24.17 Differential Equation: A differential equation is an equation that relates a function to one or more of its derivatives.

Definition 24.16 Order of a Differential Equation: The order is the highest derivative that appears in the equation.  First-order: involves only the first derivative.  
Second-order: involves the second derivative, and so on.

We can look at two examples.

(1.70)

the equation for exponential growth or decay.

(1.71)

Definition 24.17 Solution of a Differential Equation: A function y=φ(x) is a solution if substituting it into the equation makes the equation true for all x in some interval. A general solution contains arbitrary constants (one for each order of the equation). An initial-value problem (or boundary-value problem) supplies enough extra conditions to determine those constants uniquely.

Most fundamental laws tell us how something changes. Integration is the tool we use to solve many of them.

If an equation can be written in the form

(1.72)

then we separate variables and integrate both sides:

(1.73)

This is one of the most useful techniques you will learn.

For equations of the form

(1.74)

simply integrate

(1.75)

Principle 1.65:  A differential equation describes the local rule of change; its solution describes the global behavior.

Principle 1.66:  The number of arbitrary constants in the general solution equals the order of the equation.

Principle 1.67:  Initial conditions (or boundary conditions) turn a general solution into a unique particular solution that matches reality.

Principle 1.68:  Many physical laws are easier to state as differential equations than as direct relationships between quantities.

Principle 1.69:  Solving a differential equation means finding all functions that satisfy the given relation between the function and its derivatives.

Newtonian Mechanics

Newton’s laws are not merely a set of rules—they are the first great synthesis of how the universe behaves on human scales, providing the foundation upon which all of classical physics is built. Newtonian mechanics is the study of motion and the forces that produce it. It is the first complete, predictive framework in theoretical physics and remains astonishingly accurate for most phenomena we encounter in everyday life and engineering. In this review we sharpen the core ideas you will generalize later in analytical mechanics, relativity, and field theory.

Newton’s First Law

An object at rest stays at rest, and an object in motion continues in uniform motion in a straight line, unless acted upon by a net external force.

This law defines inertial reference frames and introduces the concept of natural motion (motion without force).

Newton’s Second Law

The net force acting on an object is equal to the time-derivative of the momentum

(1.76)

(for constant mass). This is the central dynamical equation of classical mechanics.

Newton’s Third Law

For every action there is an equal and opposite reaction, if object A exerts a force on object B, then B exerts a force of equal magnitude but opposite direction on A. This law is the origin of conservation of momentum in isolated systems. This is not true, in general. For example, in a spring connecting two objects it takes time for the forces to transfer through the spring.

Newton’s Equation of Motion

Combining the above, the fundamental differential equation governing a particle is

(1.77)

Solving this equation (analytically or numerically) gives the trajectory of the particle.

Work and Energy

The work done by a force along a path is

(1.78)

The Work–Energy Theorem states that the net work done on a particle equals the change in its kinetic energy

(1.79)

Principle 1.70: Inertial frames are privileged in that physics looks simplest in them. This idea becomes profoundly important when we reach special relativity.

Principle 1.71: Force is the cause of change in motion (acceleration), not motion itself. This overturned centuries of Aristotelian thinking.

Principle 1.72: Conservation laws (momentum, energy, angular momentum) emerge directly from Newton’s laws when the system is isolated or the forces have special symmetries.

Principle 1.73: Newtonian mechanics is deterministic in that given initial positions and velocities, and the forces, the future (and past) motion is completely determined. This determinism will be challenged and refined in later lessons on chaos, statistical mechanics, and quantum theory.

Algebraic Structures

Algebraic structures reveal the hidden patterns that underlie equations, symmetries, and conservation laws throughout physics. They show us that many seemingly different phenomena are governed by the same abstract rules. An algebraic structure is a set equipped with one or more operations that satisfy specific rules. Understanding these structures gives you a deeper, more unified view of mathematics and prepares you for the abstract reasoning required in gauge theories, quantum mechanics, and relativity.

Binary Operations and Closure

A binary operation on a set S is a rule that combines any two elements of S to produce another element in S. The set is closed under the operation if the result always stays inside S.

Associativity and Semigroups

An operation is associative if (a ∗ b)∗ c=a ∗(b ∗ c). A set with an associative operation is a semigroup.

Identity and Monoids

An element e is an identity if a ∗ e=e ∗ a=a for all a. A semigroup with an identity is a monoid.

Inverses and Groups

An element a has an inverse if . A monoid in which every element has an inverse is a group. Groups are among the most important algebraic structures in physics. A group that is commutative is called an abelian group.

Distributivity, Rings, and Fields

When we have two operations (usually called addition and multiplication) that interact nicely, we get richer structures:

A ring has addition (forming an abelian group) and multiplication (forming a monoid) with distributivity.

A field is a ring in which every nonzero element has a multiplicative inverse. The real numbers R and complex numbers C are fields.

Principle 1.74: Algebraic structures capture the essential rules that many different physical systems obey. Once you recognize the structure, you can transfer knowledge from one area of physics to another.

Principle 1.75: Groups describe symmetries. Noether’s theorem tells us that every continuous symmetry corresponds to a conserved quantity—one of the deepest insights in theoretical physics.

Principle 1.76: Fields provide the “numbers” we use for measurements and calculations. Their properties (commutativity, inverses, etc.) underpin everything from vector spaces to quantum operators.

Principle 1.77: Abstract algebraic thinking trains you to see the common skeleton beneath diverse physical laws. This abstraction is what allows theoretical physics to unify seemingly unrelated phenomena.

Linear Algebra

Linear algebra is the language of linearity. It gives us a precise way to describe how quantities transform, how systems are coupled, and how symmetries act—ideas that permeate every branch of theoretical physics. Linear algebra is the study of vector spaces and the linear mappings between them (we will discuss this idea in the next lesson). It is one of the most powerful and widely applicable tools in theoretical physics, underlying coordinate transformations, quantum mechanics, tensor analysis, and differential geometry.

Vector Spaces

A vector space V over a field F (often the real numbers R or complex numbers C) is a set equipped with two operations—vector addition and scalar multiplication—that satisfy the following axioms:

Closure under addition: For any u,v∈V, u+v∈V.

Commutativity: u+v=v+u.

Associativity: (u+v)+w=u+(v+w).

Additive identity: There exists a zero vector 0 such that u+0=u.

Additive inverse: For every u there exists −u such that u+(−u)=0.

Closure under scalar multiplication: For any scalar c∈F and u∈V, c u∈V.

Distributivity: c(u+v)=c u+c v and (c+d)u=c u+d u.

Compatibility: (c d)u=c(d u).

Identity: 1u=u.

Linear Independence

A set of vectors is linearly independent if the only scalars satisfying

(1.80)

are all zero.

Basis and Dimension

A basis for a vector space is a linearly independent set that spans the entire space. The number of vectors in a basis is the dimension of the space (denoted dim V).

Exercises

Exercise 1.1: Compute the derivative of using the appropriate rules. Simplify as much as possible.

Exercise 1.2: Determine whether the function is continuous at x=2. If not, classify the discontinuity and suggest a way to make it continuous.

Exercise 1.3: A car travels 120 km in 2 hours. Use the Mean Value Theorem to prove that at some instant its instantaneous speed was exactly 60 km/h.

Exercise 1.4: Use Newton’s Method with initial guess to find the positive root of to at least three decimal places. Perform at least four iterations.

Exercise 1.5: Evaluate using substitution. Then verify your result by differentiation.

Exercise 1.6: Compute using integration by parts (you may need to apply it more than once).

Exercise 1.7: Let . Find F'(x) and F''(x).

Exercise 1.8: Estimate the number of breaths you have taken in your lifetime. Clearly state your assumptions and show your reasoning.

Exercise 1.9: Given and , compute:
1)
(b)
(c) The angle between and .

Exercise 1.10: Given and , compute . Verify that the result is perpendicular to both vectors.

Exercise 1.11: A particle has position meters. Find the velocity and acceleration vectors at t=2 sec. Also compute the speed at that instant.

Exercise 1.12: Let . Compute ∂f/∂x, ∂f/∂y, and the mixed partial ∂2f/(∂y∂x).

Exercise 1.13: A 5 kg object is subject to a force F(t)=(10t,−20,0) N. Find its velocity and position at t=3 sec if it starts from rest at the origin.

Exercise 1.14: Determine whether the vectors , , are linearly independent. If not, express one as a linear combination of the others.

Exercise 1.15: Why is it more powerful to think in terms of vector spaces, groups, and linear transformations rather than just working with specific numbers and equations? Give at least two physical examples where this abstract viewpoint leads to deeper insight.

For Further Reading

Alexander Altland, Jan von Delft, (2019), Mathematics for Physicists, Cambridge University Press.

Frank Ayers Jr., Elliott Mendelson, (1999), Schaum’s Outline of Calculus, 4th Edition, McGraw-Hill Companies, Inc.

Richard Bronson, (1989), Schaum’s Outline of Theory and Problems in Matrix Operations,  McGraw-Hill Companies, Inc.

Murray R. Spiegel, (1971), Schaum’s Outline of Advanced Mathematics for Engineers and Scientists, McGraw-Hill Companies, Inc.

Leonard Susskind, George Hrabovsky, (2013), The Theoretical Minimum, Basic Books

Hermann Schulz, (2015), A Theoretical Physics Primer, Verlag-Europa-Lehrmittal

Martin M. Lipschutz, (1969), Schaum's Outline of Theory and Problems in Differential Geometry, McGraw-Hill Companies, Inc.

I. S. Sokolnikov, R. M. Redheffer, (1958), Mathematics of Physics and Modern Engineering, McGraw-Hill Book Company, Inc.

George Hrabovsky, (2026), Theoretical Physics for Amateurs vol1: Fundamentals of Math and Physics, https://www.madscitech.org/theory/index.html