Section II, Chapter Six: Mathematical Interlude
Note—the mathematics of scalars, vectors, derivatives, and integrals are more general than this chapter suggests. I’m highlighting a few of the main concepts as they pertain to Newtonian mechanics.
Note—this is a draft of Chapter Nine of a book I’m writing. The book is going to cover humanity’s deepest ideas from philosophy, physics, epistemology, and economics. Each chapter is meant to be short and digestible. Most of the chapters will explain just one or two ideas.
‘Chapter []’ indicates a future chapter that I’ve not yet written.
Note—the mathematics of scalars, vectors, derivatives, and integrals are more general than this chapter suggests. I’m highlighting a few of the main concepts as they pertain to Newtonian mechanics. For a more detailed account, see The Theoretical Minimum: What You Need to Know to Start Doing Physics by Leonard Susskind and George Hrabovsky. For a historical and qualitative account, see Infinite Powers: How Calculus Reveals the Secrets of the Universe by Steven Strogatz.
Scalars and Vectors
Consider again the algebraic expression of Newton’s Second Law: F = m*a. Mathematically, F (force) and a (acceleration) are vectors, while m (mass) is a scalar.
A scalar is “uniquely described by a real number…associated if necessary with a unit of measurement.” In the context of Newton’s Second Law, this means that we can completely specify an object’s mass by a single real number in units such as grams or kilograms in the International System of Units (SI), or in ounces or pounds in the United States customary system. Other examples of scalar quantities include volume, temperature, and energy (see Chapter []). Each such quantity is measured in distinct units, but all share the characteristic of being describable by only a single real number and corresponding units of measurement.
A vector is a slightly more complicated object. Like scalars, they require real numbers and units of measurement for specification. But vectors need three such numbers, one for each direction of space. This allows us to express a quantity’s direction, without which we cannot fully specify quantities such as force, acceleration, or velocity. For example, if a force of 10 Newtons acts on an object of 5 kg, we cannot employ F = m*a to determine the object’s resultant acceleration. We can, admittedly, determine the acceleration’s magnitude, but we’d remain ignorant of its direction in space. Acceleration due to the Earth’s gravitational field is 9.8 meters per second squared towards the center of the Earth.
Consider a projectile whose initial velocity is 10 meters per second straight upward. We may label its velocity as (0, 0, 10 m/s), where the first component represents the projectile’s velocity in the left-right direction, the second component represents the in-out direction (in and out of the page, that is), and the third component represents the up-down direction.
If the projectile’s initial velocity is instead not straight upward but is launched at an arbitrary angle relative to one’s chosen reference frame (see Chapter Six), then the above slots will in general no longer be zero.
With our updated notation for vectors, we may write out Newton’s Second Law as (Fx, Fy, Fz) = m*(ax, ay, az), where the subscripts x, y, and z correspond to the three independent directions of space.
To multiply scalars with vectors, simply multiply the scalar with each component of the vector. For example, if m = 10 kg and a = (2 m/s2, 3 m/s2, 6 m/s2), then m*a = (20 kg*m/s2, 30 kg*m/s2, 60 kg*m/s2).
Derivatives and Integrals
Newtonian mechanics deals with equations of motion: equations that tell us how an system’s trajectory evolves as a function of time. As we’ll see in Chapter [], a system’s trajectory can be expressed in terms of its position, velocity, or even acceleration over time. If we only know how one of these quantities changes with time, can we determine how the others do as well?
We can, thanks to the mathematical machinery invented by Isaac Newton and Gottfried Leibniz called calculus.
Imagine you know that a system’s velocity evolves with time such that v(t) = 9*t2 + 7t + 2 (In general, y(x) means ‘y as a function of x’. Also, we’ve abstracted away the units in this equation.).
Acceleration is defined as the instantaneous change in velocity over the change in time, or dv / dt. Without calculus, we may calculate the average acceleration between two moments in time by calculating the velocity at the beginning and endpoints of a given time interval and then dividing by the length of the interval (the duration): (v(t2) - v(t1)) / (t2 - t1) = a(t2 - t1). For the time interval t1 = 0 to t2 = 1, this yields a(t2 - t1) = 9.
But, for one thing, this is a far cry from a formula that gives us the system’s acceleration during any arbitrary time interval. And for another, this only gives us the average acceleration during a given time interval, rather than the system’s instantaneous acceleration at a singular point in time.
Note that the smaller the time interval we choose when calculating the average acceleration, the closer we get to approximating the instantaneous acceleration. For instance, we could set t1 = 0.01 to t2 = 0.01000001 and determine the average acceleration between those endpoints just as we’d done earlier, and we’d be far closer to obtaining the system’s instantaneous acceleration than when we’d set t1 = 0 and t2 = 1. But as a matter of principle, we would still not have determined instantaneous acceleration, nor would we have discovered a formula for doing so.
Calculus allows us to take the leap from mere calculating the average change along some duration to calculating then instantaneous change at an exact point in time. It gives us mathematical machinery whose input is our initial equation and whose output is a new equation (called the derivative) that tells us how the initial equation changes as a function of its dependent variable. Generically, if y(x) is some function, then the tools of calculus allow us to calculate dy(x) / dx, or how y(x) changes as a function of x.
Going back to our example in the context of Newtonian mechanics, if v(t) = 9*t2 + 7t + 2, then the rules of calculus tell us that the system’s instantaneous change in velocity over the change in time, or its acceleration, is dv / dt = a(t) = 18*t +7.
Now imagine we wish to calculate our system’s position as a function of time, x(t). Velocity is defined as a system’s change in position over the change in time, or v(t) = dx / dt. Rearranging, we have dx = v(t)*dt. We know v(t), and so we have dx = (9*t2 + 7t + 2)*dt. But we don’t want dx, which is an infinitesimally small change in x. We want x as a function of time, or x(t).
Without calculus, we would simply apply approximation techniques to deduce x(t) from v(t). For instance, we might plot v(t) in Cartesian coordinates, with t along the x-axis and v(t) along the y-axis. Then, because distance is velocity times time, then for any interval from t1 to t2, the area under the curve representing v(t) and above the x-axis gives us the exact distance traveled during that time interval. But in graphic form, v(t) happens to be a parabola, and so we cannot determine the exact area under the curve. Instead, we might approximate this area with a series of mathematically friendly rectangles. The sum of the area of the rectangles yields our approximate x(t2 - t1). The more rectangles we use, the closer our approximation is to the actual answer.
Integration may be thought of as taking the limit in which the number of rectangles goes to infinity, allowing us to calculate the exact area under the curve for an arbitrary function. As with derivatives, the machinery calculus gives us formulae for determining a function’s integral, Y(x), given an initial function y(x).
Note that because the area under the curve at any singular point is zero, we often calculate the integral of a function between two points. In our example, since v(t) = 9*t2 + 7t + 2, calculus tells us that its integral with respect to time is v(t)*dt = x(t) = 3t3 + (7/2)t2 + 2x + C, where C is a constant.
With x(t) in hand, we may calculate how much the system has traveled between any two arbitrary points in time. For instance, between t1 = 0 to t2 = 1, the system has traveled x(1) - x(0) = 3 + 7/2 + 2 + C - C = 17/2.
Summary
A scalar is a quantity that may be fully described by a real number and the requisite units. Examples include mass and time.
A vector is a quantity that may be fully described by a real number, a direction, and the requisite units. It may be expressed as a set of components, one for each direction in space. Examples include force, acceleration, and velocity.
A derivative is a function, dy(x) / dx, that tells you how the original function, y(x), changes with respect to x. For instance, if we know a system’s velocity as a function of time, then the machinery of calculus tells us how to calculate how velocity changes as a function of time, which is the system’s acceleration.
An integral is a function, Y(x), that tells you the area under the curve of the original function, y(x). For instance, if we know a system’s velocity as a function of time and plot it as a graph, then the machinery of calculus tells us how to calculate the area between the curve and the x-axis between two points t1 and t2, which is the distance that the system travels in that duration.
Will the final version of the book use ‘metre’ or ‘meter’ for the unit of length?