A High School Student's Introduction to Physics

Introduction

A car passes a speed camera and the reading says sixty miles an hour. Suppose the driver objects that she was on that road for seven minutes, so she never went sixty miles in any hour, and never could have. She has a point. The reading is a claim about one instant. An instant has no duration, and in no duration nothing travels any distance at all, so the quantity the camera reports is a distance of nothing divided by a time of nothing. Either the reading means nothing, or it means something that dividing a distance by a time cannot say.

Richard Feynman opened his introductory physics course with that difficulty, and named what it costs to resolve.

"This idea was invented by Newton and by Leibniz, independently, and is the beginning of a new branch of mathematics, called the differential calculus. Calculus was invented in order to describe motion, and its first application was to the problem of defining what is meant by going '60 miles an hour.'"

The calculus is the answer to the first question anyone asks about a moving body, and the question stays unanswerable without it. A course that means to discuss motion meets that fact in its opening minutes.

In the United States, most such courses do not. Of the more than 1.2 million students enrolled in a high school physics course of some kind in 2018 and 2019, about 79,000 sat in AP Physics C, the one course of the set built on the calculus. Six in a hundred. The other ninety-four are taught a subject called algebra-based physics, in which the derivative is withheld and the results that need it are supplied ready-made.

This article is written for one of those ninety-four.

It assumes you can add and subtract whole numbers. Everything else it uses, it builds. Multiplication, division, fractions, negative numbers, the real numbers, functions, the limit, the derivative and the integral are all constructed here, in order, with nothing borrowed forward and nothing left standing on an appeal to a course you have taken or have yet to take. Whether this is easy is a separate question, and the answer is no. The claim is that it is possible, and that what stands between a sixteen year old and the second law of motion is a timetable.

That distinction runs through everything below. The split between algebra-based and calculus-based physics names a constraint on schools, on what can be staffed and what can be sequenced across four years, and it stops there. Inside the subject the two are one thing. Newton made the calculus and the mechanics together, and neither was prerequisite to the other. An article is under no timetable at all, which is why it can decline the split.

The plan is in four parts. Part I asks what a number is, what an equation asserts, and what a measurement means, then follows a single question to its end: you can add two arrows and you can scale one, so why has nobody told you how to multiply them. The answer, forced rather than chosen, is the geometric product, and it generates the algebra in which the rest of the article works. Part II builds the calculus, defining the limit in full and the derivative through a slope function that makes the rules cheap to prove. Part III turns the machinery on motion: a constant force, a circle, a spring, a conserved quantity and an orbit, with every formula derived and its hypothesis left visible. Part IV counts what the derivative bought, and says what changes in three dimensions.

Each part opens by saying plainly what the schoolroom treatment does at the same point, and where it hides its working. Those passages quote their sources word for word, so that nothing rests on my characterisation of a book you can open yourself.

1. Counting

Begin with what you can do already. You can count, which is to say you can name the whole numbers and so on without end, and you can add two of them or subtract one from another. That is the entire starting stock. Every other operation in this article is built from it, in the order it is needed.

Two words are wanted first, because Part I turns on them. A set here means a collection of things definite enough that for anything you can say whether it belongs. The whole numbers form a set, written . An operation on a set is a rule that takes two members of it and returns a single answer.

Two pieces of shorthand go with this. When a rule takes one member of a set and returns a member of a set , write , and write to name the answer it gives. Writing for the collection of ordered pairs, one drawn from each, an operation on is a rule . Section 11 returns to this and makes the two sets carry weight; here the arrows abbreviate what has just been said in words.

The question to ask of any operation is whether the answer stays inside the set.

Addition is closed on . Add two whole numbers and a whole number comes back, every time, with no exceptions to check. Subtraction fails.

A failure of closure is an instruction. It says that the set is too small for the operation you want, and it says exactly which new members are wanted: the ones that would answer the questions currently going unanswered. Following that instruction four times occupies the next section, and it produces every kind of number this article uses.

2. The ladder

Each rung of the ladder below is forced by an equation that has no solution on the rung beneath it. Nothing is invented for its own sake, and at every stage the new numbers are exactly the answers that were missing.

The integers. The equation has no solution in , by the proposition above. Adjoin a solution to for every pair of whole numbers, call the result , and the enlarged set is the integers , on which subtraction is closed by construction. The negative numbers are not a new species of thing; they are the answers to subtractions that previously had none.

Multiplication. With in hand, define for a whole number as added to itself times, and extend it to negative by requiring that continue to hold. This is a definition, and it uses only the addition already available. Multiplication is closed on .

The rationals. The equation has no solution in , since multiplying any integer by gives an integer that is or at least in size. Adjoin a solution to for every pair of integers with , call it , and the result is the rationals . Division by anything other than zero is now closed. Division by zero is not adjoined, because for every , so the equation has no solution anywhere and no consistent answer can be supplied.

At this point every axiom of ordinary arithmetic is available. They are set down once here, so that later sections can say which of them survive.

The rationals form a field. So do the real numbers of the next paragraph. Section 6 asks which of these axioms hold for arrows, and the answer drives the rest of Part I.

The reals. One rung remains, and it is forced by a question about length rather than about arithmetic. A square with sides of length has a diagonal whose length satisfies . No rational number does.

So has holes, in the sense that a perfectly definite length fails to be named by any member of it. The real numbers are what you get by filling them. Stating precisely what filling means takes one more definition, and this article takes the standard route of characterising by the property that does the filling rather than building it piece by piece out of .

Completeness is what lacks. The set of rationals whose square is below is bounded above, by for instance, and among the rationals it has no least upper bound, since any candidate can be beaten by one slightly closer to the diagonal's length. In that least upper bound exists, and it is the number written .

Completeness is doing more work here than filling one hole. It is the property that makes the limit of Section 12 exist when it ought to, and through the limit it is what makes the derivative and the integral possible at all. Every result in Part III rests on it, at a distance of several definitions.

The ladder is now finished. Whole numbers, integers, rationals, reals, each rung forced by a question the rung below could not answer, and no rung added for any other reason. Everything numerical in the remainder of the article lives in .

3. The equals sign

Four different assertions get written with the same sign, and telling them apart is most of what algebra asks of anyone. They look identical on the page.

The distinction matters because the permitted moves differ. Both sides of an identity may be replaced by anything they equal, anywhere the identity is stated to hold. A constraint may be manipulated only in ways that preserve its solution set, so multiplying both sides by an expression that might vanish can introduce solutions the original never had. A law may be rearranged as an identity once assumed, and its content lies entirely in the assumption.

A great deal of confusion in a physics class comes from meeting the fourth kind while trained only on the third. A student who has spent two years solving for reads as an instruction to find something, when it is a statement that the world behaves in a certain way, from which things may then be found.

4. Measurement

A measurement is a number against a unit. The number alone says nothing: three is not a length, and three metres is. Writing a quantity means writing both, and the pair travels together through every calculation.

Units combine under multiplication and division exactly as numbers do. A distance divided by a time gives metres per second, and dividing that by a time again gives metres per second per second. Nothing here needs a new rule. What does need stating is the restriction that addition obeys.

The proof says something worth keeping in mind. A sum of unlike quantities fails to be a quantity because its numerical value would depend on which units you happened to choose, and a physical statement cannot depend on that. Dimensional homogeneity is a consequence of the world not caring what units anyone uses.

The same shape recurs later. Quantities carry a label. Multiplication combines the labels according to a rule. Addition refuses to mix quantities with different labels, and the refusal is not a matter of taste; the sum would fail to mean anything. Section 9 attaches a different label, called grade, to the elements of an algebra of arrows, and the same three sentences will be true of it. A reader who is comfortable with the fact that a length cannot be added to a mass has already met the only idea needed to accept that a number cannot be added to an area.

5. Where things are

Physics is about things that are somewhere, so before any law can be written the space of positions has to be pinned down. The difficulty is that physical space, as anyone experiences it, has no special point in it. A stone falls the same way in Caracas and in Edinburgh, and no experiment picks out an origin. Any faithful model has to start without one.

That definition withholds a great deal. It gives differences of positions and nothing else. There is no origin, and there is no way to add two points, because the sum of two places is not a place. This is faithful to the physics, in which separations and displacements are measurable and an absolute location never is.

Two small consequences follow immediately and get used without comment later.

Now comes the move that lets the whole article proceed without any further machinery. Choose a point and keep it.

The identification [eq:position-vector] costs a choice, and the choice is arbitrary: a different origin gives every point a different position vector. It buys the right to stop distinguishing a place from the arrow that reaches it, so that a moving body becomes a single arrow-valued function of time, and every derivative taken in Part III lands back in the same space where the positions live. Displacements are unaffected by the choice, since [eq:affine-chain] makes the displacement from to equal to whatever origin was picked, which is the only sense in which positions may be subtracted.

Everything now depends on , which has been named but not described. That is the next section.

6. Directed quantities

A displacement carries a length and a direction, and the objects that do so are what this article calls arrows. Two operations on them need no justification, because both are forced by what displacements are. Arrows add, by laying one after the other, and [eq:affine-chain] is that addition. Arrows scale by real numbers, since half of a displacement is a definite displacement and so is its reverse.

A set of vectors is linearly independent when the only way to combine them to is with every coefficient zero, and a basis is a linearly independent set whose combinations exhaust . Every basis of a given space has the same number of members, and that number is the dimension of the space. This word was used differently in Section 4, where the dimension of a quantity was its list of unit exponents; the two senses are unrelated and the context always settles which is meant. This article works throughout in dimension two, the plane, with a chosen basis written , so that every vector is for exactly one pair of reals.

Now compare [eq:vector-axioms] with the field axioms [eq:field-axioms] of Section 2, and audit the arrows against the numbers.

That is the gap, and it is a strange one to have been left with. School spends years establishing that numbers can be added, subtracted, multiplied and divided, then introduces a second kind of quantity, teaches two of the four operations, and moves on. A reader who noticed the omission and asked about it was probably told that arrows are simply not the sort of thing you multiply.

They are. The remaining sections of Part I do it, and the product turns out to be forced rather than chosen. Before that, look at the two things school offers in place of a product. Understanding why neither is a multiplication is most of the way to finding the one that is.

7. The two half-answers

The dot product measures how much of one arrow lies along another, and it is useful; Section 25 builds work and energy from it. As a candidate for multiplication it fails at the first test of Section 1. It is not closed: feed it two arrows and a number comes back. Multiplying twice is impossible, since asks for the dot product of a number with an arrow, which is undefined. An operation you cannot apply twice is not an arithmetic.

The other offer is worse.

Three separate objections apply, and the article will return to each.

The first is that it does not exist here. This article works in the plane, and the definition asks for a direction perpendicular to two independent directions.

So in the plane the cross product returns nothing at all. This is not a technicality to be waved away by working in three dimensions instead. Part III shows that a planet orbiting in a plane has a perfectly definite angular momentum, and that a spinning wheel in a plane has a perfectly definite torque, neither of which needs a third direction to exist in. Whatever those quantities are, the cross product is not what they are; it is something three dimensions happen to allow you to use in their place.

The second objection is that even where it exists, it is not closed in the sense that matters. The hand rule is a convention. Reflect the whole configuration in a mirror and every arrow reflects with it, while comes back pointing the wrong way, because the mirror image of a right hand is a left one. A quantity whose direction depends on a convention chosen by the person doing the calculation is a bookkeeping device.

The third objection is that neither offer is a multiplication at all, and school presents them as though between them they cover the ground. One returns a number and throws the plane away. The other returns an arrow and throws the magnitude of the plane into a direction that only exists in three dimensions. Each keeps half of what the pair of arrows knows.

There is a reason to press on this rather than accept it. David Hestenes, who spent a career on both physics teaching and the algebra this article is heading towards, put the consequence bluntly.

"perhaps a third of the students seem incapable of reasoning with vectors as abstract elements of a linear space. Rather, they insist on conceiving a vector as a list of numbers or coordinates. I have come to regard this concept of vector as a kind of conceptual virus"

He is describing graduate students. The infection starts early, and it starts here, with an algebra so incomplete that coordinates are the only way to compute in it. The next section closes the gap, and it does so by asking for one thing and taking whatever follows.

8. The geometric product

We ask for a product of arrows that behaves like multiplication in every structural respect, and we ask for one thing more, chosen because it is the only sensible thing to ask. Everything else is then determined. Nothing is invented, nothing is chosen for convenience, and the object school calls a bivector arrives whether we want it or not.

The first demand asks only that the thing deserve the name multiplication. The second carries the content, and it is the natural request to make. An arrow is not a number, so a product of arrows cannot be expected to be a number in general. There is exactly one number an arrow carries on its own, its length, and the one product whose value is beyond argument is the product of an arrow with itself. Demand [eq:contraction] says that squaring an arrow gives the one number it already knows. That is the entire input.

The dot product was never assumed. It was extracted, from a demand that mentions only lengths. Whatever else the geometric product does, half of it is already familiar.

The other half has no choice about it either. Every product splits into a part unchanged by swapping the factors and a part that changes sign, since

and this is an identity, true by inspection. The first bracket has just been identified. Name the second.

The number in [eq:wedge-coefficient] is the signed area of the parallelogram the two arrows span, positive when the turn from to runs one way and negative when it runs the other. So carries exactly the information the cross product was trying to deliver, an oriented area, and it carries it without appealing to a third dimension or to a hand.

Equation [eq:geometric-product] adds a number to an area, which looks wrong. Section 4 already answered it.

"You have probably noticed that the expanded multivector form (30) violates one of the basic math strictures that is drilled into our students, namely, that 'it is meaningless to add scalars to vectors,' not to mention bivectors and pseudoscalars. On the contrary, GA tells us that such addition is not only geometrically meaningful, it is essential to simplify and unify the language of physics"

9. The algebra of the plane

The demands of Section 8 have consequences that can be written out completely, because in the plane the whole algebra is small enough to fit on one line.

Equation [eq:i-squared] retires something that has probably been presented to you as a mystery. The square root of minus one is not an imaginary quantity smuggled in to make equations work. It is the oriented unit area of the plane you are standing in, and the reason it squares to minus one is [eq:basis-relations], which says nothing more than that two perpendicular unit vectors anticommute.

Everything the algebra contains can now be listed.

Compare this with the closing paragraph of Section 4. Quantities carry a label. Multiplication combines the labels by a rule, since a vector times a vector gives grades zero and two, as [eq:geometric-product] says. Addition holds different labels apart without mixing them. The bookkeeping that a reader already accepts for metres and kilograms is the bookkeeping that runs this algebra, and is how you read one entry off the list.

The multiplication table follows from [eq:basis-relations] and [eq:i-squared] alone.

One consequence deserves separating out, because it explains where the complex numbers came from.

That is exactly the multiplication rule for complex numbers, with in the place of . The complex numbers are the even part of the algebra of the plane, and they were never a separate invention. Section 10 shows what they do.

10. Rotors and reflections

The usual way to write what follows is with an exponential, , and that requires an infinite series, which requires the limit of Section 12. This article does not borrow forward, so everything below uses the algebra already in hand, and Section 17 supplies the exponential notation once the limit exists to support it.

The condition says the pair lies on the unit circle, so the rotors form a circle's worth of objects, one for each direction you might turn through.

Now attach the usual names. Write for the rotor whose circle parameters are the coordinates of the point reached by travelling a distance anticlockwise around the unit circle from , and call those coordinates and . That the arc length is a well-defined number is a fact about integration, supplied in Section 17; until then is a label for a point on the circle and nothing below depends on more than that.

With those names, [eq:rotor-compose] reads

which are the angle-addition formulas, obtained here as a statement about multiplying two elements of an algebra. Anyone who has memorised them will recognise what has happened: they are the rule for composing turns, and composing turns is multiplying rotors.

Two further facts finish the section. The first is a dimensional accident, and this article flags such things rather than letting them pass as general truths.

The two-sided form is the law that holds in every dimension, and in three dimensions the one-sided form is simply false, because there the unit volume commutes with vectors rather than anticommuting with them. In the plane the two coincide, so this article may use the shorter throughout while knowing that the general statement is [eq:sandwich]. Section 28 says what changes.

The second fact explains where turns come from.

Part I is finished. Starting from arrows that could be added and scaled and nothing else, one demand produced a closed multiplication, and out of it came an oriented area, the complex numbers, the trigonometric addition formulas, and the algebra of turns and reflections. Part II builds the calculus, and Part III puts the two together.

Part II builds the calculus, which is the thing the algebra-based course exists to avoid. Before building it, the case for avoiding it deserves stating at full strength, because it is better than its reputation and because the article's answer depends on having heard it.

Stewart Brekke taught physics and chemistry in Chicago public schools from 1975 to 2001, mostly in schools where a majority of students were Black or Hispanic. He wrote to Physics Today in 2021 arguing for algebra-based physics, and he began from a fact that ought to reframe the whole dispute.

"In most US public high schools, only the upper 35% of the student body is allowed or encouraged to enroll in algebra-based physics classes. Such restrictions often exclude many otherwise bright and capable youngsters—of all races and ethnicities—from that formative first step toward a future in physics."

The quarrel over whether the calculus belongs is being conducted above a gate that two thirds of students never reach. Before anyone is asked to choose between two kinds of physics course, most are offered neither.

Brekke's second fact concerns who teaches. He cites a national survey of about 27,000 US high school physics teachers, which found that 32 per cent held a degree in physics or physics education, and that four in ten teach a majority of their classes in other subjects. His own restatement, that two thirds of US high schools lack such a teacher, shifts the unit from teachers to schools and the survey does not establish it, so the survey's own figure is the one to carry.

From those facts Brekke concludes that the course should stay algebra-based, and reports his method: prepared formula sets, and about ten problems per formula, so that repetition through physics teaches the algebra.

The facts are real and the conclusion does not follow from them. That most physics teachers lack a physics degree is an argument about what teaching materials must carry, and it says nothing about how much mathematics a curriculum should contain. Brekke half concedes this, since his own remedy is a materials remedy; he chose thin materials and then treated the ceiling as fixed. Thin staffing argues for better materials. It never argues for less mathematics.

Subject knowledge is not dispensable. Section 20 quotes research whose entire method is diagnosing why a student's wrong answer is wrong, and telling a productive confusion from a plain error takes expertise no worksheet supplies. A teacher without it can hand a student this article. They cannot repair a student who has misread it.

The strongest version of the access argument is more recent and more careful. Suzanne White Brahmia and Geraldine Cochran put it this way.

"In the U.S., introductory calculus-based courses often acts as a gatekeeper to STEM degrees. But access to the prerequisite math is far from equal—students from low-income and racially marginalized communities are far less likely to have the chance to take calculus before college. These disparities are then compounded by rigid placement systems that reward procedural fluency in algebra and trigonometry, while overlooking the conceptual quantitative reasoning that physics truly demands. Labeling students as 'underprepared' ignores the structural causes of these gaps and places the burden of remediation on those least supported."

Every word of that holds, and what they propose matters most. Their remedy is an extended, credit-bearing course that embeds quantitative reasoning and gives students more room, in their phrase to "expand access without compromising rigor". The strongest published argument for access does not conclude that the calculus should go. It concludes that the room should grow. An article is under no timetable, and a reader may take a year over this one.

A word about where I am standing, since the argument is partly about students like the one I was. I came to the United States from Venezuela and went through its public schools. I sorted myself into the advanced track and stayed there, taking more than sixteen Advanced Placement courses across two Houston public high schools, and I did not find them hard. What I found was that difficulty, as school used the word, almost always meant procedural volume: longer computations, more of them, arranged to fill a timetable I was legally required to sit inside. Very little of it asked a question worth the time.

So I balk when the case for a thinner physics course is made on behalf of students like me. The proposal takes a subject that has been emptied of its reasoning and offers it to the students with the least access to anything better, then calls that access. Underrepresented students are owed the subject itself. Handing them a contrived version and calling it enriching is the condescension, wearing the language of help.

There is also a question of what the prerequisite is protecting. A study of 150 Calculus I final examinations from across US colleges and universities classified 3735 individual items and found that 85.21 per cent could be answered by recalling a fact or applying a memorised procedure, with the examinations seldom asking for explanation and rarely requiring a student to use the central ideas of the course at all. Whatever a calculus prerequisite certifies, it is largely technique.

And technique does not travel. The finding that ought to end the argument is nearly forty years old.

"Frequently students who have no trouble plotting points and computing slopes cannot apply what they have learned about graphs from their study of mathematics to physics."

Students who can compute a slope perfectly well cannot read a slope as a rate. The mathematics course delivered the procedure and the procedure did not transfer, which is what happens to procedures learned without the ideas beneath them. Part II builds those ideas. It defines the limit in full, defines the derivative in a way that makes its rules provable in two lines each, and constructs the integral, and it assumes nothing except Part I.

11. Function

Three parts of that definition matter. The domain is part of the function, so on the non-zero reals and the same formula on the positive reals are different functions. Each input gets exactly one output, which is what makes a name for something definite. And nothing requires the rule to be a formula; a function is an assignment, and formulas are one way of describing assignments.

12. The limit

The camera in the introduction reported a speed at an instant, and dividing a distance by a time could not produce one. The way out is to ask what the ratio approaches as the interval shrinks, and to make "approaches" mean something exact. This section does that, and it is the most demanding definition in the article.

The idea is a challenge and a response. Someone claims that approaches as approaches . To test the claim you name a tolerance: get within of , for whatever small you like. The claim survives if for every tolerance you name, the claimant can name a nearness to that delivers it.

The clause excludes deliberately. The limit asks where the function is heading as the input closes in, and it must be able to ask that even where the function has no value at all, which is exactly the situation the speed camera creates.

Every proof from [eq:limit-def] has that shape. The tolerance arrives first and the nearness is built in response, with the response allowed to depend on the tolerance. Reversing the order would be a different and much weaker claim.

Two facts about limits are used constantly below.

13. Continuity

All three clauses do work. A function can fail to be continuous by having no value at the point, by having no limit there, or by having both and having them disagree. The third case is the one that looks like nothing is wrong.

Section 14 defines the derivative by building a function of exactly this shape and then insisting the value at the point be the right one. Repairing a hole by supplying the limit as the value is the mechanism.

14. The derivative

Return to the speed camera. Over an interval from to the average speed is the distance divided by the time, and the quantity we want is what that ratio becomes when the interval closes up. Written out, the ratio is

which is a perfectly good number for every except , where it reads and means nothing.

Section 13 met a function with a hole and repaired it by supplying the limit as the value. The definition below performs the same repair, and it does so by refusing to divide in the first place.

Away from the equation forces to be the ratio [eq:difference-quotient], so records the slope of the chord from to . At itself the equation says only and imposes nothing, and the entire content of the definition is the demand that be continuous there. Continuity is what ties the value at the point to the values around it.

One example shows the mechanism.

No limit was taken and no appeared. The factor was cancelled before it could cause trouble, which is legitimate because the cancellation happens away from and continuity carries the result in.

The same trick handles every power at once, with no binomial expansion.

The definition owes a debt to the one every other book uses, and paying it takes one theorem.

So nothing has been given up. The two definitions describe the same functions and assign the same derivatives. What has been gained appears in the next section, where every rule of differentiation becomes two lines of algebra with no limit manipulation at all.

Part III uses the following without comment.

15. The rules

Each rule below follows the same recipe: write the difference of the outputs, extract a factor of , and read off what multiplies it. The function that multiplies it is the slope function, and its value at is the derivative.

The next one is the reason for the whole approach. In the usual treatment the chain rule is the hardest of the three, because dividing by is illegitimate whenever that difference vanishes, and the standard proofs work around it. Here the difficulty does not arise, because nothing is ever divided.

Three rules, three proofs, each a short piece of algebra ending in a continuity remark. That is what the definition was chosen to buy, and Section 17 collects the second dividend, when the same argument goes through unchanged for functions whose values are multivectors rather than numbers.

16. The integral

The derivative answers what a quantity is doing at an instant. The integral answers the reverse question: given the rate at every instant, what is the total. Section 20 needs it to turn an acceleration into a position, Section 25 needs it to turn a force into an energy, and Section 17 needs it to say what an angle is.

The picture is the familiar one of rectangles under a curve, with and squeezing the area from below and above. The definition avoids saying what area means and instead says the two approximations agree, which is a statement about numbers alone.

One theorem about derivatives is still owed, and the fundamental theorem below cannot be proved without it.

That closes the debt Section 10 recorded. The rotor parameters and are now coordinates of a point located by a definite number, and Section 17 can differentiate them.

17. Differentiating multivectors

Part III studies bodies whose position is a vector varying with time, so the calculus of Part II has to survive the change from numbers to multivectors. It survives untouched, for the following reason.

[eq:multivector-caratheodory] is [eq:caratheodory] with no change at all. The factor is a real number, and real numbers commute with everything in the algebra, so no question of ordering arises and the definition needs no repair. Differentiating componentwise gives the same answer, since each component equation is an instance of [eq:caratheodory].

The product rule does change, in exactly one respect.

Writing in the wrong order would be wrong, and a familiar consequence fails: , which is only when commutes with its own derivative. Section 21 uses this.

What remains is the derivative of the rotor, and one limit does all the work.

Sections 21 and 22 rest on [eq:rotor-second]. Something turning at a steady rate has an acceleration equal to times its position, and circular motion, centripetal acceleration and the harmonic oscillator all follow from that with no further idea.

The order in which a subject is taught is often mistaken for the order in which it must be learned. Physics is presented as coming after the calculus, so it looks as though the calculus is a prerequisite. The history says otherwise, and it says so in a way that bears directly on what a course can leave out.

Newton had the calculus before he wrote the Principia. He then wrote the Principia without it, in the geometrical style of the ancients, and the scholarship on why has run since the 1690s.

"This question comes very naturally to mind, since Newton discovered the calculus of fluxions before writing the Principia. It is just obvious to think that Newton had employed the calculus in order to mathematize his natural philosophy. However, very little trace of calculus techniques is to be found in the Principia, which are mostly written in 'geometric style'."

Guicciardini argues at length that the geometry of the Principia is not merely calculus in disguise, and that the mathematicians who later set out to translate it into calculus found real difficulties. Only the ordering is claimed here. Newton did not complete a course in the calculus and then take up mechanics. He made both, and the sequence taught today is a decision about timetables.

Newton said himself why the Principia looks as it does, writing in the late 1710s about readers who had by then changed.

"To the mathematicians of the present century, however, versed almost wholly in algebra as they are, this synthetic style of writing is less pleasing, whether because it may seem too prolix and too akin to the method of the ancients, or because it is less revealing of the manner of discovery. And certainly I could have written analytically what I had found out analytically with less effort than it took me to compose it."

He found the results analytically and then rewrote them geometrically, at greater cost to himself, for an audience he judged unready. That is the algebra-based physics course, described three centuries early by the person whose name is on it.

It cost him something specific. The geometry available to him could not express the very quantity his second law is about.

"the geometric mathematics Newton used in the Principia — and others were using before him — had no way of representing acceleration as a quantity in its own right." … "But the geometric mathematics used in the Principia offered no way of representing second derivatives."

Newton worked around it by using the curvature of the path, the circle that touches a curve, wherever a second derivative was wanted. Section 21 reaches for a second derivative directly and gets the same result in two lines.

One more constraint on this Part explains a choice made back in Section 3.

"most of the mechanics problems in introductory physics are 2D problems … Conventional vector algebra cannot do this, in part because the vector cross product is defined only in 3D. That is the main reason why coordinate methods dominate introductory physics. The available math tools are too weak to do otherwise."

Everything below happens in the plane, where the cross product does not exist and nothing is lost by its absence.

18. Kinematics

Both derivatives exist as objects of the same kind as the position, because differentiating a vector-valued function of one real variable returns a vector, and the identification of a point with an arrow is what permits this. Without it, a velocity would have to live somewhere other than where positions live, and the machinery for saying where would be the theory of manifolds.

The speed at an instant, which the introduction claimed was unavailable to arithmetic, is now simply . Nothing is divided by zero, because nothing is divided.

19. The second law

Recall the four uses of the equals sign from Section 3. Equation [eq:newton-two] is the fourth kind. It asserts something about the world that could have been false, and its content is that the force determines the second derivative of the position and nothing else. It says nothing directly about where the body is, or how fast it is going. Give it a force law, a starting position and a starting velocity, and everything else follows by integration.

That is the method of Part III, and the remaining sections are five instances of it.

20. Constant force

Take the simplest case, a force that does not change. A dropped stone near the ground is the standard example, and [eq:newton-two] gives , a constant vector which we call .

Those three formulas, together with the two that follow from eliminating between them, are the set a physics student is asked to memorise. They are a corollary here, obtained by integrating a constant twice, and every one of them carries its hypothesis on its face: the acceleration is constant. A spring, a planet, a pendulum and a falling body meeting air all violate it, and for those the formulas say nothing at all.

School makes one further claim alongside them.

[eq:trapezoid] is the trapezoid rule, which approximates the area under a graph by the area under the chord. It is exact for a straight graph and wrong for every other, and [eq:trapezoid-error] is the size of the error.

Now compare that with how the result is presented. The most widely used free textbook in American classrooms, OpenStax College Physics, states the formula and marks its hypothesis correctly, writing "(constant a)" beside it. That is more careful than it is often given credit for. The justification it offers is the whole difficulty:

"The equation […] reflects the fact that, when acceleration is constant, v is just the simple average of the initial and final velocities."

Read as a statement it is true. Read as a reason it is empty, because "the simple average" is exactly what was to be shown. Proving it takes the definition [eq:average-velocity] and the fundamental theorem, and those are the two things an algebra-based course does not have. So the kinematic formulas, which the book derives correctly from that starting point, rest on a base that cannot be established within the course that teaches them.

The consequence shows up in what students can do afterwards. The research is nearly forty years old and has been replicated many times since.

21. Circular motion

A body going round a circle of radius at a steady rate has a position that is a fixed vector turned through an angle growing steadily with time. Section 10 says what turning is and Section 17 says how to differentiate it, so the trajectory can be written down at once:

with the constant rate of turning. Everything about the motion now follows by differentiating.

That is circular motion. Two differentiations of one line, and the result that directed inwards arrives with no geometry and no approximation anywhere in the argument.

Set the standard derivation beside it. OpenStax College Physics obtains the same result from a picture of two similar triangles.

"Using the properties of two similar triangles, we obtain Δv/v = Δs/r." … "Acceleration is Δv/Δt, and so we first solve this expression for Δv: Δv = (v/r)Δs. Then we divide this by Δt, yielding Δv/Δt = (v/r) × (Δs/Δt). Finally, noting that Δv/Δt = a_c and that Δs/Δt = v … the magnitude of the centripetal acceleration is a_c = v²/r." … "(Because Δθ is very small, the arc length Δs is equal to the chord length Δr for small time differences.)"

Three of those steps are false as written, and each is false in the same way.

is not an equality. The ratio is an average acceleration over an interval, and is its value at an instant; the two agree only in a limit. is not an equality, for the same reason. And an arc is never equal to its chord for any interval of positive length, however small; the two approach a common ratio, which is a statement about a limit and not about equality.

The word limit does not appear anywhere in that passage. What has happened is that a derivative has been taken three times, correctly, and each time the taking has been recorded as an equals sign. The result is right because the errors vanish in a limit that is never mentioned. The reasoning cannot be followed, because as written it is invalid, and a student who takes it at face value has learned that mathematics is a procedure that emits correct answers for reasons one is not invited to examine.

Section 3 separated four uses of the equals sign. This passage adds a fifth, in which the sign means "becomes, once something we are not going to discuss has been done".

22. The spring

A spring pulls back in proportion to how far it is stretched, which is Hooke's law: the force is for a positive constant , with the minus sign recording that the pull opposes the displacement. Feed that into [eq:newton-two]:

Compare [eq:oscillator] with [eq:centripetal]. They are the same equation, with . A mass on a spring and a body going round a circle are governed by one statement, and the section is therefore already finished.

The figure traces the component, because a horizontal connector makes the projection visible. That component is the sine, which is the same motion started a quarter turn earlier and satisfies the same equation.

So the cosine is a theorem. It is one component of a rotor, and the reason the oscillation is sinusoidal is that turning at a steady rate is what solves [eq:oscillator].

The same result reaches a student in the algebra-based course like this.

"The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke's law, is given by x(t) = X cos(2πt/T)"

There is no derivation, because there cannot be one. Equation [eq:oscillator] relates a function to its own second derivative, and a course without derivatives has no way to write it down, let alone solve it. The answer is therefore handed over, and the student is left to take on trust the single most important fact about oscillation, which is that it is circular motion seen from the side.

That fact is usually mentioned as an analogy. It is [eq:shm-solution].

23. Splitting the law

Section 8 built a product of vectors with two parts, a number and an area. Every use of it so far has been in one piece. This section takes it apart, because the two parts of one product turn out to be the two conservation laws of mechanics.

Multiply the second law [eq:newton-two] by the position, using the geometric product, and split by [eq:geometric-product]:

Do the same with the velocity:

Two products, four pieces, and the two that matter are the bivector part of the first and the scalar part of the second. This section takes the bivector, and Section 25 takes the scalar.

Torque is an oriented area, of grade two, and it is a perfectly definite object in the plane. Compare that with the treatment a student meets, in which the torque has a magnitude and a direction supplied by holding up a hand.

24. Angular momentum

That is a conservation law, and it took three lines. What produced it was a differentiation and a term that vanished because a wedge of parallel things is zero, with no appeal to symmetry, to experiment, or to the inverse-square law. It holds for any central force.

The physical meaning of follows from the meaning of the wedge.

Kepler found that law by fitting twenty years of observations. Here it is a corollary of the conservation law above, and the derivation never mentions gravity or ellipses. It holds for a planet, for a mass on a string being whirled in a circle, and for a charged particle attracted to a fixed charge.

The treatment a student receives states the law and then says, accurately, what it does not supply.

"Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times." … "The point is to demonstrate that the force of gravity is the cause for Kepler’s laws (although we will only derive the third one)."

So the second law is left where Kepler left it, as a regularity extracted from twenty years of observation. Its cause is [eq:L-dot] with the right-hand side equal to zero, which is three lines from the second law of motion and unavailable to a course that cannot differentiate a product.

25. Work and energy

Section 23 split two products and set one piece aside. This section takes the scalar part of [eq:vF-split], the quantity , and finds the other conservation law of mechanics inside it.

Equation [eq:work] is an ordinary integral of an ordinary function of , so Section 16 already covers it. The usual notation abbreviates the same thing.

So the work and energy theorem is the fundamental theorem of calculus, applied along a trajectory. A quantity was differentiated and the total of the derivative recovered the change in the quantity.

Some forces admit a second bookkeeping.

Two cases cover everything in this article.

The quantity is a formula students are given. Here it is the result of an integration, and the integration is the reason the one half is there. A course without the integral must supply the one half as a fact to be remembered, alongside the in the kinetic energy and the in [eq:suvat], as three separate things. They are the same thing three times: the integral of a linear function.

26. One product, two conservation laws

Everything is now in place to say what the article has been building towards.

Part I asked for a product of arrows and found that one demand determined it. The product had two parts, a number and an oriented area, and Section 7 had shown that school offers each part separately while pretending between them they cover the ground. Keeping both is worth the following.

Equation [eq:summit] is one statement. There is a single product of vectors. It has a scalar part and a bivector part. Contracted against the velocity, its scalar part is the rate of change of energy. Contracted against the position, its bivector part is the rate of change of angular momentum. The two great conserved quantities of mechanics are the two grades of one multiplication, and the conditions under which each is conserved are the two ways a product can lose a grade: the dot vanishes when the factors are perpendicular, the wedge vanishes when they are parallel.

A circular orbit under gravity manages both at once. The force is central, so and the angular momentum holds. The force is also perpendicular to the velocity, so and the speed holds. Both facts, which are usually presented as separate observations about orbits, are the two grades of and vanishing for the same reason: the force lies along one vector and across the other.

This is what the dot product and the cross product were unable to say. Each of them keeps one grade and discards the other, so each can state one conservation law and is blind to the second. Keeping the product whole states both, and shows them to be the same statement viewed at two grades.

27. The tally

Four results were examined in the form a student meets them.

Now count what each route costs a student to carry.

The algebra-based route requires five kinematic formulas together with the condition restricting them to constant acceleration, a memorised , a memorised sinusoid, a memorised period, a right-hand rule, a separate statement of each conservation law, and Kepler's laws as three facts about planets. Each arrives on its own authority, and none can be checked from anything else on the list.

The route taken here requires one law of motion, [eq:newton-two]; one product of vectors, [eq:geometric-product]; and two operations of the calculus that are inverse to one another. Everything else in Part III was derived, and each derivation is short enough to redo from memory.

The second list is shorter than the first. That is the argument of this article. It does not claim that the calculus is easy, or that Part II can be read quickly. It claims that the total quantity of material a student must hold is smaller when the material is connected, and that the algebra-based course is longer to learn because a list of unrelated facts is harder to carry than a short argument.

There is a second cost, harder to count. Every derivation above stated its hypotheses and used them. Where a result held only under a condition, the condition appeared in the statement, and Section 20 could say exactly how wrong the average-velocity formula becomes when its hypothesis fails, which is [eq:trapezoid-error]. A course that supplies results ready-made cannot do this, so its results carry no conditions, and a student has no way to know when one of them stops applying. Knowing when a formula fails is most of what it means to understand it.

One last thing about the arrangement of this article. The order was: number, then equation, then measurement, then position, then the product of arrows, then function, limit, derivative, integral, then motion. At no point was anything used before it was built, and the audit that checks this is part of the article's construction rather than a claim made about it.

So a reader who has arrived here did not need a prerequisite. There was never one to have.

28. Three dimensions

Everything above happened in the plane. Most of it transfers without alteration. The parts that change are named below, so that the plane is not mistaken for the general case.

The construction of Section 8 never used the dimension. One demand, , gives [eq:polarisation] and hence the split [eq:geometric-product] in any number of dimensions. What changes is what the algebra contains.

In three dimensions there are three independent planes, spanned by , and , so bivectors form a three-dimensional family rather than the one-dimensional family of Section 9. There is also a grade-three object, the oriented unit volume , and the algebra has eight dimensions in total.

Two facts about the plane were flagged as accidents when they were used, and both now fail.

The first is that anticommutes with vectors, which is what made rotation one-sided in Theorem [eq:sandwich]. In three dimensions commutes with everything, and the one-sided formula is simply wrong. The two-sided sandwich is the law that holds everywhere, and the plane's shorter form is a coincidence of the case where the unit area happens to anticommute.

The second is the cross product, and here is where it comes from. In three dimensions, and only there, the bivectors and the vectors both form three-dimensional families, so the two can be matched up. The matching is multiplication by the unit volume:

The cross product is therefore the wedge, converted into a vector by an operation available only because three happens to equal three. In the plane there is nothing to convert to, which is Proposition [eq:no-cross-in-plane], and in four dimensions bivectors form a six-dimensional family, so there is far too much to convert to. The right-hand rule is the record of a choice of sign in [eq:cross-dual], and its bad behaviour under reflection is the sign of changing when the orientation does.

Everything in Part III survives the move. The second law is unchanged, angular momentum stays the bivector , torque stays , and Theorem [eq:summit] holds word for word. What a reader of the three-dimensional treatment gains is rigid bodies, which need the extra planes, and gyroscopes, which are unavailable in the plane because a plane has only one axis to spin about.

That development is the subject of Classical Mechanics from Zero, in Two Languages, which builds the same algebra over a general inner-product space, carries the matrix and geometric languages in parallel, and reaches rigid-body dynamics and the Lagrangian and Hamiltonian formalisms. The linear algebra underneath, developed as a subject in its own right, is Linear Algebra I.

A reader who has followed this article to here has met, in order and with nothing assumed: the construction of the number systems, the axioms of a field and of a vector space, the geometric product and the algebra it generates, the limit, the derivative, the mean value theorem, the Riemann integral and the fundamental theorem, and five laws of motion derived rather than quoted. That is a first course in mathematics and a first course in physics, and it is one course, because they were never two.