Before You Begin: The Mathematics Physics Assumes

School physics is a collection of formulas. University physics is a small set of principles, expressed in vectors and calculus, from which the formulas are derived. The transition fails not because the physics gets harder but because the language changes: a statement like ”$\mathbf{F} = m\mathbf{a}$ where $\mathbf{a} = \ddot{\mathbf{r}}$” assumes you can read vectors, differentiate, and treat an equation between vector quantities as three equations at once.

This page rebuilds that language from first principles; vectors, trigonometry, coordinate geometry, and calculus; defining every term before it is used and proving every rule. A confident reader should take the self-test in §5 and only study the sections whose answers surprise.

Physics quantities come in two kinds. A scalar; mass, energy, temperature - is a number. A vector; displacement, velocity, force; is a number and a direction. The single most important upgrade from school physics is learning to compute with directions: a car turning a corner at constant speed is accelerating, because its velocity vector is changing. Nothing in school algebra can say that sentence; vectors can.

The second upgrade is calculus. Speed is the derivative of position; acceleration is its second derivative. Every equation of motion; the ones school gives you as $s = ut + \frac{1}{2}at^2$; is a differential equation, and the school formulas are what you get by solving it for constant acceleration. University physics derives them; this page gives you the differentiation that derivation needs.

Definition. A vector $\mathbf{v}$ is a quantity with magnitude and direction, written $|\mathbf{v}|$ for its magnitude. Two vectors are equal when their magnitudes and directions agree.

Components. In a plane, $\mathbf{v} = v_x\,\hat{\mathbf{i}} + v_y\,\hat{\mathbf{j}}$ where $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ are unit vectors along the axes and

These four identities resolve every “resolve the force along the slope” exercise, and they are the definition of component rather than a trick to memorise.

Addition by components. $(u_x + v_x,\; u_y + v_y)$; the parallelogram rule from school is component addition, proved by applying each vector’s definition.

Scalar multiplication. $c\,\mathbf{v}$ has magnitude $|c||\mathbf{v}|$, direction preserved for $c > 0$ and reversed for $c < 0$. Force balance $\sum\mathbf{F} = 0$ is a statement about component sums, and it is why a book on a table has weight balanced by a normal force of equal magnitude and opposite direction.

The dot product.

Proof of the equivalence. Both sides are bilinear in the components, so it suffices to check the two unit vectors along the axes: $\hat{\mathbf{i}}\cdot\hat{\mathbf{i}} = 1 = 1\cdot1\cos 0$ and $\hat{\mathbf{i}}\cdot\hat{\mathbf{j}} = 0 = \cos 90°$; bilinearity then extends it to every pair. $\blacksquare$

The dot product answers “how much of $\mathbf{u}$ points along $\mathbf{v}$” - which is the question every projection, every work integral, and every component-of-gravity-along-a-slope calculation is asking.

Definition (radian). An angle $\theta$ radians subtends arc length $\theta$ on the unit circle. All calculus below assumes radians.

The identities, with proofs rather than a list.

• $\sin^2\theta + \cos^2\theta = 1$: the equation of the unit circle.
• $\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$: obtained by rotating the point $(\cos\alpha, \sin\alpha)$ through $\beta$.
• $\sin 2\theta = 2\sin\theta\cos\theta$ and $\cos 2\theta = 1 - 2\sin^2\theta$: the addition formulas at $\alpha = \beta$.

The two limits every physics derivation uses:

They are why $\frac{d}{dx}\sin x = \cos x$ in radians; and why that derivative is false in degrees. If you have ever differentiated $\sin$ in degrees and gotten a wrong trajectory, this is why.

Small-angle physics. $\sin\theta \approx \theta$ and $\cos\theta \approx 1 - \theta^2/2$ for small $\theta$, which is the linearisation that makes pendulums harmonic and lenses computable; and the second-order term is what makes the correction, so a pendulum’s period grows with amplitude rather than staying constant.

Definition. The gradient of the line through $(x_1,y_1)$, $(x_2,y_2)$ is $m = \frac{y_1 - y_2}{x_1 - x_2}$; two lines are perpendicular iff $m_1 m_2 = -1$.

Proof of perpendicularity. Direction vectors $(1, m_1)$ and $(1, m_2)$ are perpendicular iff their dot product vanishes: $1 + m_1 m_2 = 0$. $\blacksquare$

Straight-line graphs as linear models. $y = mx + c$ is the statement that the response changes at a constant rate $m$ per unit of the input. Every “equation of the form $y = mx + c$” exercise in school is a linear model, and reading $m$ and $c$ off a graph is reading a derivative and an initial condition.

Areas and the quadratic. Projectile range $R = \frac{v^2\sin 2\theta}{g}$ is maximised at $\theta = 45°$ because $\sin 2\theta$ peaks at $1$ when $2\theta = 90°$; a trigonometry fact doing the work of a differentiation, and a preview of how the university course will use both.

Definition. $f'(x) = \lim_{h\to 0}\frac{f(x+h) - f(x)}{h}$ when the limit exists; geometrically the gradient of the tangent.

The table, with the two proofs that matter.

Proof for $x^r$ from first principles. Expand $(x+h)^r$ by the binomial theorem, subtract $x^r$, divide by $h$; every surviving term carries one factor $h$, so the limit is $r x^{r-1}$. $\blacksquare$

Proof for $\sin$. From the addition formula, $\frac{\sin(x+h)-\sin x}{h} = \sin x\,\frac{\cos h - 1}{h} + \cos x\,\frac{\sin h}{h} \to \cos x$, using the two limits of §2. $\blacksquare$

Rules. Product: $(fg)' = f'g + fg'$. Chain: $(g \circ f)' = g'(f)\,f'$. The chain rule is the one that makes physics work: if $x$ is a function of $t$ and $v = \frac{dx}{dt}$, then $\frac{dv}{dt} = \frac{dv}{dx}\cdot\frac{dx}{dt} = v\,\frac{dv}{dx}$, which converts every acceleration-into-position problem into a first-order separable equation.

Integration. $\int_a^b f\,dx$ is the signed area under the curve, and the fundamental theorem of calculus says it is computed by antiderivatives: $\int_a^b f = F(b) - F(a)$. The work–energy theorem $W = \int \mathbf{F}\cdot d\mathbf{r} = \Delta(\tfrac12 mv^2)$ is that theorem applied to Newton’s second law.

Differential equations, in one move. The statement

is the simple harmonic motion equation, and $x = A\cos(\omega t + \phi)$ is its solution; verifiable by differentiating twice. Verification is a legitimate solution method at this level, and it is how the school formulas are recovered from the principles.

1. A force of $50\,\mathrm{N}$ acts at $37°$ above the horizontal. Give its horizontal and vertical components. (§1)
2. Two forces $\mathbf{F}_1 = (3, 4)$ and $\mathbf{F}_2 = (-4, 3)$ act on a body. Find the resultant’s magnitude. (§1)
3. What is $\cos 2\theta$ in terms of $\sin\theta$ only? Prove it. (§2)
4. Why is $\frac{d}{dx}\sin x = \cos x$ false in degrees? (§2)
5. Show that the lines $y = 3x + 1$ and $x + 3y = 7$ are perpendicular. (§3)
6. Differentiate $f(x) = x^2\sin x$. (§4)
7. Verify that $x = A\cos(\omega t)$ solves $\ddot{x} = -\omega^2 x$. (§4)
8. A ball is thrown at $20\,\mathrm{m/s}$ at $30°$. Find its range using the double-angle identity, then state the angle that maximises it. (§2, §3)

Answers. (1) $(50\cos 37°, 50\sin 37°) \approx (39.9, 30.1)\,\mathrm{N}$. (2) $(3-4,\, 4+3) = (-1, 7)$, magnitude $\sqrt{50} \approx 7.07$. (3) $\cos 2\theta = 1 - 2\sin^2\theta$, from $\cos(\theta+\theta)$ with the addition formula. (4) Because $\frac{\sin h}{h} \to \frac{\pi}{180}$ when $h$ is in degrees, so the derivative picks up that factor. (5) Gradients $3$ and $-\tfrac13$; product $-1$. (6) $2x\sin x + x^2\cos x$ by the product rule. (7) $\dot{x} = -A\omega\sin(\omega t)$, $\ddot{x} = -A\omega^2\cos(\omega t) = -\omega^2 x$. (8) $R = \frac{400\sin 60°}{9.81} \approx 35.3\,\mathrm{m}$; maximised at $\theta = 45°$.

• Postulates of Quantum Mechanics; vectors again, but in Hilbert space.
• Newtonian Mechanics Review; track 1, which begins where this page ends.
• The Laws of Thermodynamics; the scalars-only track, for contrast.

• About This Site
• Complete University Physics Study Guide
• Physics
• Physics Glossary, Key Terms and Definitions