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A LITTLE CLARITY GOES A LONG WAY

Big ideas.
Clear foundations.

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FIELD NOTES / MATHEMATICSy = x²xy0
01 / PATTERNS INTO UNDERSTANDING↗

START WITH THE FUNDAMENTALS

Four subjects. Plenty to discover.

Short explanations, essential equations,
and examples you can work through.

01Algebra ↘02Calculus ↘03Physics ↘04Chemistry ↘

01 / MATHEMATICS

Algebra

Describe relationships. Find the unknown.

Solving a linear equation

An equation states that two expressions are equal. To isolate a variable, perform the same operation on both sides so the equality remains true.

WORKED EXAMPLE

Solve 3x + 7 = 22

  1. Subtract 7 from both sides: 3x = 15.
  2. Divide both sides by 3: x = 5.
  3. Check: 3(5) + 7 = 22. The solution works.

Read the slope

In y = mx + b, m is the slope and b is the y-intercept. For y = 2x + 3, increasing x by 1 increases y by 2; the line crosses the y-axis at 3.

Try it: Solve 4x − 6 = 18.

Show answer

Add 6, then divide by 4. x = 6.

02 / MATHEMATICS

Calculus

Understand change and accumulation.

The derivative: an instantaneous rate

A derivative describes how quickly a function changes at a point. Geometrically, it is the slope of the tangent line to a curve.

WORKED EXAMPLE

Differentiate f(x) = x³

The power rule gives d(xⁿ)/dx = nxⁿ⁻¹. Therefore f′(x) = 3x². At x = 2, the slope is 3(2²) = 12.

The integral: adding small contributions

A definite integral measures net accumulation over an interval. For a nonnegative function, it gives the area under the curve. For example, ∫₀² x dx = [x²/2]₀² = 2.

Try it: What is the derivative of 5x²?

Show answer

Using the constant multiple and power rules, the derivative is 10x.

03 / NATURAL SCIENCE

Physics

Connect measurements to models.

Force, mass, and acceleration

Newton’s second law relates the net force on an object to its mass and acceleration: Fnet = ma. Force is measured in newtons (N), mass in kilograms (kg), and acceleration in metres per second squared (m/s²).

WORKED EXAMPLE

A 2 kg cart experiences a net force of 6 N

Rearrange the equation: a = Fnet/m = 6/2 = 3 m/s². The acceleration points in the direction of the net force.

Always check the units

Units help you catch mistakes. A newton is 1 kg·m/s², so dividing force by mass leaves m/s², the correct unit for acceleration. Remember that net force includes all forces acting on the object.

Try it: What net force accelerates a 4 kg object at 2 m/s²?

Show answer

F = ma = 4 × 2 = 8 N.

04 / NATURAL SCIENCE

Chemistry

Explore matter, amounts, and reactions.

From grams to moles

The mole counts entities such as atoms or molecules. One mole contains exactly 6.02214076 × 10²³ entities. Molar mass connects an amount in moles to a mass in grams: n = m/M.

WORKED EXAMPLE

How many moles are in 18.0 g of water?

Using a molar mass of approximately 18.015 g/mol for H₂O: n = 18.0/18.015 ≈ 0.999 mol (three significant figures).

Balance atoms, not subscripts

A balanced chemical equation conserves each element’s atoms. In 2H₂ + O₂ → 2H₂O, each side has four hydrogen atoms and two oxygen atoms. Change coefficients to balance an equation; changing subscripts changes the substances.

Try it: Balance N₂ + H₂ → NH₃.

Show answer

N₂ + 3H₂ → 2NH₃. Both sides contain two nitrogen atoms and six hydrogen atoms.

MAKE YOUR STUDY TIME COUNT

Read. Work it out.
Explain it back.

01 — Start with the idea.
Write what a formula means before substituting numbers.

02 — Show each step.
Track units, signs, and assumptions as you work.

03 — Test your understanding.
Try an example without looking, then explain why it works.

A place to start.

College Math & Science Hub is an independent educational website with free introductory notes for learners reviewing college-level fundamentals. These short guides support self-study and are not a complete course.

We are not affiliated with a college or university and do not award credit or qualifications. For coursework, follow your instructor’s notation, assumptions, and required materials.