Mathematics and Physics: Why They Should Be Studied Together

Mathematics is the language of physics: derivatives are velocity and acceleration, integrals are work and displacement, vectors are forces and torques, differential equations are Newton's, Maxwell's and every other law. Understanding this correspondence is not a philosophical detail: it's what separates the student who applies formulas from memory from the one who reasons. Newton and Leibniz invented calculus precisely to describe motion, and since then every advance in physics has also been an advance in the mathematics used to express it. Studying them together makes both easier -- and more interesting.

A scene that recurs in many classrooms. The student is good at physics: solves problems, remembers formulas, gets decent grades. Then the chapter changes, simple harmonic motion arrives, and everything is harder. The teacher introduces second derivatives, and the student suddenly feels they no longer "understand physics". The problem isn't the physics -- it's that physics has become so entangled with mathematics that without seeing the link, the two worlds feel like two unrelated subjects, when in fact they are two sides of the same coin.

This guide is for those who want to see the link explicitly: the correspondence between mathematical and physical concepts, why it exists, how to use it in daily study. It is not a theoretical article -- it's a practical tool to study better, valid both in high school and in the first year of Engineering.

In this guide:

• A story: Newton, Leibniz and calculus to describe motion
• The fundamental correspondence: derivative and velocity
• Integral and work: the other half of calculus
• Vectors: the key to forces
• Trigonometry: inclined planes and the harmonic oscillator
• Differential equations: how the laws of nature are written
• Studying them together: the method that works
• School curricula: what changes with the 2024 Guidelines
• FAQ

A story: Newton, Leibniz and calculus to describe motion

Between 1660 and 1680, two mathematician-physicists working independently -- Isaac Newton in England and Gottfried Leibniz in Germany -- developed what we now call differential and integral calculus. Newton called it the "method of fluxions"; Leibniz introduced the dx/dt notation we still use today. But the crucial point, for our purposes, is the reason they invented it: not to do abstract mathematics, but to describe the motion of bodies.

Newton wrote the Philosophiae Naturalis Principia Mathematica in 1687, and in that book he established the laws of motion and universal gravitation using -- implicitly -- the calculus he had developed in the preceding years. The law we now write as F = m·a or, more precisely, F = m·d²x/dt², was not an invention separate from mathematics: it was an identity between a physical concept (force) and a mathematical operation (the second derivative of position with respect to time).

Nearly two centuries later, James Clerk Maxwell published in 1865 his A Dynamical Theory of the Electromagnetic Field, in which the equations of electromagnetism were expressed as partial differential equations. The result: theoretical prediction of electromagnetic waves, confirmed later by Hertz, and the whole telecommunications technology that followed. Physics could make that leap only because mathematics had provided the appropriate language.

Since then the story has repeated: relativity and differential geometry, quantum mechanics and functional analysis, string theory and advanced algebraic geometry. Every great advance in physics has also been an advance in the mathematics used to write it. Mathematics is the language of physics is not a metaphor -- it's a historical observation.

The fundamental correspondence: derivative and velocity

The derivative of a function measures how rapidly it changes. In physics, the quintessential changing quantity is the position of a body in time, and its rate of change has a name everyone knows: velocity. The correspondence is not by analogy -- it is identical.

Formally:

• Instantaneous velocity = dx/dt, derivative of position with respect to time.
• Instantaneous acceleration = dv/dt = d²x/dt², derivative of velocity (and second derivative of position).

In high school this correspondence is seen only from year 4 or 5 onwards, when derivatives are introduced in analysis. Before that, you work with average velocities (Δx/Δt) and average accelerations (Δv/Δt). But the difference between "average" and "instantaneous" is exactly the difference between incremental ratio and limit, that is, between algebra and calculus. The derivative allows you to talk about "velocity at an instant" -- a fundamental physical concept that without calculus would not even be rigorously definable.

Concrete example. A body moves with law x(t) = 3t² + 2t. What is its velocity at t = 2 s? Without derivatives, you would have to compute average velocities over ever smaller intervals and trust the limit. With derivatives, dx/dt = 6t + 2, so v(2) = 14 m/s. Direct. The uniformly accelerated motion of high school (s = s₀ + v₀·t + ½·a·t²) is not a formula dropped from the sky: it's the result of integrating constant a to get v, and v to get s.

To consolidate kinematics and see this connection in action, the natural starting point is our guide to high school mechanics: kinematics and dynamics.

Integral and work: the other half of calculus

If the derivative answers "how fast is this quantity changing?", the integral answers the opposite question: "what is the total effect of this quantity accumulated over time or space?". And that question has a direct physical translation.

Key correspondences:

• Work of a force along a displacement: W = ∫F·ds. It is an integral: sum of "infinitesimal pieces of work" (F·ds) along the path. For constant force and rectilinear displacement, the integral reduces to the high school product F·s, but only as a special case.
• Displacement from velocity: Δx = ∫v·dt. If velocity varies in time, displacement is no longer "velocity times time" but the area under the v-t graph.
• Impulse of a force: J = ∫F·dt = Δp. The impulse-momentum theorem (force times time = change of momentum) is an integral.
• Kinetic energy as integral of work: the work-energy theorem links the integral of F along the displacement to the change of ½·m·v² -- a result rigorously proven only with calculus.

Here too, high school introduces integrals in year 5, and suddenly many things in physics done in previous years acquire new meaning. The work of a variable force on an inclined plane? Integral. The energy stored in a spring? Integral of F = k·x. The volume of water displaced in a variable-section pipe? Integral. Physics becomes more powerful because it can finally handle non-linear cases.

Vectors: the key to forces

Almost all errors in dynamics come from one thing: treating vectors as numbers. Force, velocity, acceleration, momentum are vector quantities -- they have magnitude, direction and sense. Adding two forces is never adding two numbers: it's a vector sum, with the parallelogram rule or decomposition along axes.

Vector operations and their physical counterparts:

Vector decomposition along axes is the most powerful high school tool: it turns a "2D" problem into two "1D" problems on the chosen axes. On an inclined plane, decomposing weight into P·sin θ (parallel to the plane) and P·cos θ (perpendicular) makes the problem solvable with F = m·a on each axis -- without decomposition, it is unmanageable.

For students who studied vectors geometrically in high school but encounter them again at university in algebraic form (components, bases, scalar and vector product in coordinates), the transition is less traumatic if the correspondence with forces is already clear: mathematics acquires meaning because it describes something concrete.

Trigonometry: inclined planes and the harmonic oscillator

Sine and cosine come from geometry, but in physics they become universal tools for two classes of problems: decomposition of oblique quantities and oscillatory motion.

Decomposition: the weight on an inclined plane of angle θ has parallel component P·sin θ and perpendicular component P·cos θ. A force F at 30° to the motion does work F·s·cos 30°. The trigonometry of the right triangle, applied to vectors, is everywhere. Without it, inclined planes are a mystery; with it, they are routine.

Oscillatory motion: simple harmonic motion -- pendulum, spring, body oscillation -- is described by x(t) = A·cos(ωt + φ). Velocity and acceleration follow by differentiation: v(t) = -A·ω·sin(ωt + φ), a(t) = -A·ω²·cos(ωt + φ) = -ω²·x(t). The relation a = -ω²·x is a second-order differential equation, and every time physics finds a system obeying this equation, it knows it will behave as a harmonic oscillator -- vibrating strings, LC circuits, pendula, atoms vibrating in a crystal. Trigonometry is the mathematics of oscillations.

And here's the point: the same equation, a = -ω²·x, describes completely different physical systems. It is an example of the so-called "unreasonable effectiveness of mathematics in physics": a single mathematical structure unites phenomena that at first sight have nothing in common. Understanding the mathematics means seeing these deep similarities.

Differential equations: how the laws of nature are written

We arrive at the fundamental conceptual node. Almost all the laws of physics are differential equations: relations linking a quantity to its derivatives. Recognising this structure is the key to moving from high school to university physics -- and to understanding why mathematics is so central.

Examples of physical laws written as differential equations:

• Newton's second law: m·d²x/dt² = F. A second-order differential equation, in which the "driving" F can depend on position, velocity, time. Solving it yields the motion x(t) -- any motion, in principle.
• Radioactive decay: dN/dt = -λ·N, where N(t) is the number of nuclei. Solution: N(t) = N₀·e^(-λt), exponential decay.
• Charging a capacitor: dq/dt = (V - q/C)/R, first-order differential equation; solution: charge growing exponentially towards V·C.
• Wave on a string: ∂²u/∂t² = v²·∂²u/∂x², wave equation, in partial derivatives. Solutions: waves propagating with speed v.
• Schrödinger equation: iℏ·∂ψ/∂t = -(ℏ²/2m)·∇²ψ + V·ψ. The whole of quantum mechanics is enclosed in this differential equation.

In high school this level of formalization isn't reached, but the pieces can be glimpsed (Newton's law written as F = m·a, exponential decay, the harmonic oscillator). In the first year of Engineering, the tool is explicit: Analisi Matematica 1 and 2 provide the language, and Fisica I uses it from day one -- see our guide to first-year physics at the Politecnico.

Summary table of the main correspondences:

Studying them together: the method that works

Given the correspondence, effective study of physics requires parallel work on mathematics -- and vice versa. The method, distilled from experience with many students, breaks down into a few key moves.

1. When you study a new mathematical concept, look for a physical application. Just done derivatives? Think about velocity and acceleration. Introduced integrals? Think about work and displacement. Mathematics becomes much more memorable when attached to a concrete meaning, instead of remaining a manipulative abstraction.

2. When you study a physical concept, rewrite it in its full mathematical form. Studied F = m·a? Rewrite it as m·d²x/dt² = F. Studied work W = F·s? Rewrite it as W = ∫F·ds. The formalism is not an ornament, it is the "true" form of the law behind the school version.

3. Use mathematics to understand physics, not just to compute it. A typical exercise: given a motion x(t), ask yourself what kind of motion it is. If it's a linear function of t, it's uniform motion. If quadratic, uniformly accelerated. If sinusoidal, oscillatory. The mathematical form of the function tells you the nature of the motion, before doing any calculation.

4. Always draw. A v-t graph, a force decomposition, a vector representation: drawing is the natural bridge between mathematics and physics. Drawing forces you to choose axes, signs, directions -- that is, to do the conceptual work that calculus then formalizes.

5. Study in cycles, not in compartments. "I'll do mathematics first and then physics" is a losing strategy, because the two worlds reinforce each other. Half an hour of derivatives, half an hour of kinematics with derivatives, half an hour of mixed exercises, half an hour of theory on both. Short alternating cycles.

In our mathematics tutoring and physics tutoring we often work on a dual track for exactly this reason: in many cases, an apparent block in physics stems from a gap in mathematics (or vice versa), and keeping the two subjects together in study is the most efficient way to unblock.

School curricula: what changes with the 2024 Guidelines

An interesting note for high school students and families. The new National Guidelines for the Licei, published by the Italian Ministry of Education and Merit in 2024, explicitly emphasise the connection between mathematics and the physical world. The text mentions the need to show how "mathematics developed also in constant dialogue with the other sciences" and to value "the link with the history of ideas".

This is consistent with the structure of the liceo scientifico programme:

• In year 4, derivatives are introduced, and at the same time uniformly accelerated motion and dynamics with forces are studied.
• In year 5, integrals are introduced, and at the same time work, energy, waves and circuits are studied -- all contexts in which integrals appear naturally.
• First- and second-order differential equations are treated in year 5, with explicit applications to physics (harmonic oscillator, radioactive decay, RC circuits).

In practice, the liceo scientifico is already structured to teach mathematics and physics as two sides of the same conceptual tool. What often is missing, in the student's experience, is that this connection be made explicit. Two different subjects, two different teachers, two different classrooms -- and the bridge risks never being built. Working with a tutor who explicitly shows the correspondences, especially in the transition between year 4 and year 5, is often the difference between "surviving year 5" and "really understanding physics in time for first-year Engineering".

FAQ

Why are mathematics and physics studied together in the school programme? Because mathematics is the formal language of physics: derivatives describe velocity and acceleration, integrals describe work and displacement, vectors describe forces, differential equations express the laws of nature. Historically, Newton and Leibniz developed calculus (1660-1680) precisely to describe the motion of bodies. Studying them together makes both more understandable.

Can I be good at physics without being good at mathematics? Only up to a point. In the first two years of high school it's possible to get by with basic mathematics; in years 4 and 5, where physics becomes formalized with derivatives and integrals, weak mathematics becomes a serious obstacle. At university, especially at the Politecnico, the Analisi prerequisite is explicit: without solid mathematical bases, physics becomes much harder than it needs to be.

What level of mathematics does high school physics need? High school physics requires: algebra, equations and inequalities, vectors, basic trigonometry. In years 4 and 5, derivatives and integrals come in and become essential for the more advanced chapters (work and energy, oscillatory motion, electromagnetic induction). First- and second-order differential equations are explicitly used in year 5.

What does it mean that "F = m·a is a differential equation"? It means that, in full form, it is m·d²x/dt² = F. The force F can depend on position, velocity or time, and the equation links a function (position x(t)) to its second derivative. Solving it gives how the body moves in time. It's differential because it contains derivatives of the unknown.

What is the most important thing to grasp to use mathematics well in physics? Probably vectors. Many errors in dynamics arise from treating forces and velocities as numbers instead of vectors. Vector decomposition along axes is the most powerful high school tool: it turns a 2D problem into two 1D problems solvable separately. Without mastering this, every inclined plane becomes a nightmare.

Are there pure physicists who use little mathematics? At university and research level, no. Every branch of modern physics -- mechanics, electromagnetism, thermodynamics, quantum, relativity -- is expressed in mathematical language, and advanced levels require advanced mathematics (functional analysis, differential geometry, group theory). At high school level, it's possible to get decent grades with little mathematics, but you don't really understand the subject.

Is it worth doing mathematics and physics tutoring together? Often yes, especially in years 4 and 5 of high school and in the first year of Engineering. The two subjects reinforce each other: parallel work allows you to unblock apparent difficulties in physics that are in fact mathematical gaps, and vice versa. The time saving at the same result is significant.

Continue reading:

• High School Mechanics: Kinematics and Dynamics
• TOLC-I Physics: Topics and What to Study
• First-Year Physics at the Politecnico di Milano
• Physics Tutoring in Milan
• Mathematics Tutoring in Milan