Physics · Foundations · Chapter notes
Basic Maths for Physics · Calculus, Vectors and Graphs
The maths physics assumes you already have: derivatives for velocity, integrals for work and displacement, dot and cross products, graph reading, and the small-angle and binomial approximations.
In short
Physics quietly assumes a set of maths that is never taught inside a physics chapter. Derivatives give you velocity and acceleration from position, integrals give you displacement and work back, vectors handle direction, and two approximations, small-angle and binomial, turn otherwise impossible expressions into ones you can evaluate.
Contents
Why Physics Needs This Maths
Physics is written in the language of rates and sums: how fast something changes, and how many small pieces add up. That language is calculus, together with vectors and graphs. This note builds the exact toolkit JEE physics assumes you already have, so the physics itself never gets stuck on the maths.
• Differentiation for rates of change (velocity, acceleration).
• Integration for accumulation (displacement from velocity, work from force).
• Vectors for quantities with direction (force, field, momentum).
• Graphs and approximations for reading and simplifying physical relationships.
Differentiation: The Rate of Change
The derivative dy/dx measures how fast y changes as x changes. In physics it turns position into velocity and velocity into acceleration.
d/dx (sin x) = cos x · d/dx (cos x) = −sin x
d/dx (eₓ) = eₓ · d/dx (ln x) = 1/x
• Constant multiple: d/dx(cf) = c f'.
• Sum: differentiate term by term.
• Product rule: (uv)' = u'v + uv'.
• Chain rule: for a function of a function, dy/dx = (dy/du)(du/dx). The chain rule is the one physics leans on most, for example when a quantity depends on time through position.
If position x = 5t³ − 2t, then velocity v = dx/dt = 15t² − 2, and acceleration a = dv/dt = 30t. Differentiating once gives the rate; differentiating again gives the rate of the rate.
At a maximum or minimum the derivative is zero: dy/dx = 0. This locates the peak of a projectile, the maximum range, or the minimum of a potential energy curve. Check the second derivative for the type: negative means a maximum, positive means a minimum.
Integration: Adding Up the Pieces
Integration is the reverse of differentiation and the tool for accumulation: adding infinitely many infinitesimal contributions. In physics it recovers displacement from velocity and work from a varying force.
∫ sin x dx = −cos x + C · ∫ cos x dx = sin x + C
∫ (1/x) dx = ln x + C · ∫ eₓ dx = eₓ + C
A definite integral from a to b gives the area under the curve between those limits. This is exactly why the area under a velocity-time graph is the displacement, and the area under a force-displacement graph is the work done.
If v = 6t, the displacement from t = 0 to t = 2 is ∫₀² 6t dt = [3t²]₀² = 3(4) − 0 = 12 m. Integration accumulates the velocity over time into a distance.
A spring force F = −kx does work W = ∫₀ᵣ (−kx) dx = −½kx². The integral handles the fact that the force changes continuously with position, which a simple F×d cannot.
Vectors for Physics
Many physical quantities have direction as well as size. Vectors let you add, subtract and multiply them correctly.
• Dot (scalar) product: A · B = |A||B|cosθ, a number. Used for work (W = F · d) and for finding the angle between vectors.
• Cross (vector) product: A × B = |A||B|sinθ n̂, a vector perpendicular to both. Used for torque (τ = r × F) and angular momentum. The dot product peaks when vectors are parallel; the cross product peaks when they are perpendicular.
A force F = 3î + 4ĵ N moves an object through d = 2î m. Work = F · d = (3)(2) + (4)(0) = 6 J. Only the component of force along the displacement does work.
Graphs in Physics
Reading and drawing graphs is half of physics problem solving. The slope and the area both carry physical meaning.
• The slope of a graph is the derivative of the y-quantity with respect to the x-quantity.
• The area under a graph is the integral.
So for a velocity-time graph, slope = acceleration and area = displacement. For a force-position graph, area = work. Recognising which is which turns many problems into a single reading.
A straight line through the origin is direct proportion (y = mx). A parabola is a square law (y ∝ x²), like distance under constant acceleration. A rectangular hyperbola (y ∝ 1/x) appears in Boyle's law and inverse-square forces. Recognising the shape tells you the relationship at a glance.
Approximations and Angles
Physics constantly simplifies. Two tools, the small-angle approximation and the binomial approximation, turn ugly expressions into workable ones.
For a small angle θ in radians: sinθ ≈ θ, tanθ ≈ θ, and cosθ ≈ 1. This is what makes the simple pendulum simple and the thin-lens and small-oscillation formulae work. Always convert to radians first: sin(0.05 rad) ≈ 0.04998, almost exactly 0.05.
For small x: (1 + x)ⁿ ≈ 1 + nx. This linearises expressions like (1 + v²/c²) in relativity or (1 + h/R)⁻² in gravitation, where the correction is tiny. Keeping only the first-order term is standard practice.
sinθ ≈ tanθ ≈ θ, cosθ ≈ 1 (small θ)
(1 + x)ⁿ ≈ 1 + nx (small x)
The acceleration due to gravity at small height h above the surface: g' = g(1 + h/R)⁻² ≈ g(1 − 2h/R). The binomial approximation turns an inverse-square expression into a simple linear drop, valid because h is far smaller than R.
Maths for Physics: One-Page Formula Sheet
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