Nowadays students rarely read textbooks; they prefer watching videos. Well, I have good news. Strogatz taught a course based on his book, and his lectures are posted on YouTube. You can learn chaosstraight from the horse’s mouth. You better get started: there are over 24 hours of video (all embedded below).
Before you plunge into Quick Calculus, perhaps we ought to tell you what it is supposed to do. Quick Calculus should teach you the elementary techniques of differential and integral calculus with a minimum of wasted effort on your part; it is designed for you to study by yourself. Since the best way for anyone to learn calculus is to work problems, we have included many problems in this book. You will always see the solution to your problem as soon as you have finished it, and what you do next will depend on your answer. A correct answer generally sends you to new material, while an incorrect answer sends you to further explanations and perhaps another problem.
The book covers nearly all the calculus needed in IPMB.
Chapter Two discusses differentiation—including the product rule and the chain rule—and maximum/minimum problems.
Chapter Three analyzes integration, both definite and indefinite, and covers techniques such as change of variable, integration by parts, and multiple integrals.
Chapter Four summarizes all the results in a few pages.
Math Books Useful for IPMB.
The only calculus in IPMB that Quick Calculus doesn’t teach is vector calculus; for that you should consult Div, Grad, Curl and All That. Used Math covers more ground than Quick Calculus, but it’s a handbook rather than a self-teaching guide.
Quick Calculus has several virtues. It is clearly written, it emphasizes understanding math visually with lots of plots, and it focuses on utilitarian techniques without distracting rigor. If you want to understand math at a fundamental level, you should take a real calculus class. If you want to brush up on what's needed to get through IPMB, use Quick Calculus.
One disadvantage is that Quick Calculus is old. The second edition—the most recent one I am aware of—was published in 1985. It might be difficult to purchase, although Amazon seems to have copies for sale. The authors make quaint comments about “readers who have an electronic calculator,” as opposed to slide rules I suppose. I also found several typos, which might frustrate readers using the book for self-study.
A sample from Quick Calculus.
The format is unusual. The text is divided into approximately half-page “frames,” and the reader is guided from one frame to the next. Someone should put this book online, because it would lend itself to an interactive online format. Rather than explain how the book is organized, I’ve taken Section 1.17 of IPMB and rewritten it in the style of Quick Calculus (see below). In my opinion, if all of Intermediate Physics for Medicine and Biology were organized like this it would be tedious. What do you think?
Most of my reading is nonfiction, but recently I read Jonathan Swift’s Gulliver’s Travels. The story describes Englishman Lemuel Gulliver’s journeys to exotic lands, including Lilliput inhabited by tiny people, and Brobdingnag where giants live. Swift was a delightful and funny writer, but Florence Moog claims “Gulliver was a Bad Biologist” (Scientific American, Volume 179, November 1948, Pages 52–55). The problem is scaling, which Russ Hobbie and I discuss in Chapter 2 of Intermediate Physics for Medicine and Biology. The properties of animals change as they get bigger or smaller; you can’t just scale people up or down and expect them to function correctly. As Moog writes “for a student of comparative biology Gulliver’s book may serve as an unpremeditated textbook on biological absurdities.”
“Gulliver was a Bad Biologist,” by Florence Moog.
Moog’s first example was the 60-foot tall Brobdingnagians. She notes that because their mass increases as the cube of their height, supporting their body would “necessitate a truly ponderous skeleton” (A point I’ve discussed before in this blog when contemplating elephants). The giants would need thick stubby legs and fat bones.
Title Page of Gulliver’s Travels.
Moog then considers the six-inch-tall Lilliputians. “If the Brobdingnagians were too big to exist, the mouse-sized Lilliputians were too small to be human.” She explains that smaller animals have a higher specific metabolic rate (that is, rate per unit mass) than larger animals. “Gulliver … failed to realize that the creatures of his invention would have spent the larger part of their time stuffing themselves with food.”
Neil deGrasse Tyson discussing Gulliver’s Travels.
Maybe Tyson has a point. Moog concludes that “after all, we must not be too hard on Gulliver for failing to understand the biological conditions that made him a man—and an implausible liar. His talents … were in the psychological realm.” His satirical story provides great insight into human behavior.
Question: Are there neurological manifestations associated with reports of audible and sensory phenomena among US government personnel in Havana, Cuba?
Findings: In this case series of 21 individuals exposed to directional audible and sensory phenomena, a constellation of acute and persistent signs and symptoms were identified, in the absence of an associated history of blunt head trauma. Following exposure, patients experienced cognitive, vestibular, and oculomotor dysfunction, along with auditory symptoms, sleep abnormalities, and headache.
Meaning: The unique circumstances of these patients and the consistency of the clinical manifestations raised concern for a novel mechanism of a possible acquired brain injury from a directional exposure of undetermined etiology.
Overall … the JAMA paper is pretty weak. Clearly, something has happened to make these 21 people experience so many unpleasant symptoms, but at present I don’t think we can rule out the possibility that the cause is psychological in nature.
Last week’s New York Times article was triggered by the recently proposed hypothesis that microwaves are responsible for these health issues. Russ Hobbie and I discuss the biological effects of electric and magnetic fields in Section 9.10 of IPMB. We focus on the potential of microwaves to induce tumors, and conclude that nonthermal mechanisms are implausible. In other words, radiofrequency fields can heat tissue—just like in your microwave oven—but they don’t cause cancer. The hypothesis touted in the Times article, however, is a thermal mechanism: a thermoelastic pressure wave sensed as sound by part of the inner ear called the cochlea.
Hearing induced by microwaves has been studied for years, and is known as the “Frey effect” after Allen Frey, who first reported it. A 2007 article in the journal Health Physics by James Lin and Zhangwei Wang (Volume 92, Pages 621-628) describes this phenomenon.
The hearing of microwave pulses is a unique exception to the airborne or bone-conducted sound energy normally encountered in human auditory perception. The hearing apparatus commonly responds to airborne or bone-conducted acoustic or sound pressure waves in the audible frequency range. But the hearing of microwave pulses involves electromagnetic waves whose frequency ranges from hundreds of MHz to tens of GHz. Since electromagnetic waves (e.g., light) are seen but not heard, the report of auditory perception of microwave pulses was at once astonishing and intriguing. Moreover, it stood in sharp contrast to the responses associated with continuous-wave microwave radiation. Experimental and theoretical studies have shown that the microwave auditory phenomenon does not arise from an interaction of microwave pulses directly with the auditory nerves or neurons along the auditory neurophysiological pathways of the central nervous system. Instead, the microwave pulse, upon absorption by soft tissues in the head, launches a thermoelastic wave of acoustic pressure that travels by bone conduction to the inner ear. There, it activates the cochlear receptors via the same process involved for normal hearing. Aside from tissue heating, microwave auditory effect is the most widely accepted biological effect of microwave radiation with a known mechanism of interaction: the thermoelastic theory. The phenomenon, mechanism, power requirement, pressure amplitude, and auditory thresholds of microwave hearing are discussed in this paper. A specific emphasis is placed on human exposures to wireless communication fields and magnetic resonance imaging (MRI) coils.
Their introduction gives some useful numbers.
The microwave auditory phenomenon or microwave hearing
effect pertains to the hearing of short-pulse, modulated
microwave energy at high peak power by humans
and laboratory animals (Frey 1961, 1962; Guy et al.1975a, b; Lin 1978, 1980, 2004). The effect can arise, for
example, at an incident energy density threshold of 400
mJ m-2 for a single, 10-µs-wide pulse of 2,450 MHz
microwave energy, incident on the head of a human
subject (Guy et al. 1975a, b; Lin 1978). It has been
shown to occur at a specific absorption rate (SAR)
threshold of 1.6 kW kg-1 for a single 10-µs-wide pulse
of 2,450 MHz microwave energy. A single microwave
pulse can be perceived as an acoustic click or knocking
sound, and a train of microwave pulses to the head can be
sensed as an audible tune, with a pitch corresponding to
the pulse repetition rate (Lin 1978).
The temperature increase caused by such a microwave pulse is rapid (microseconds) and tiny (microdegrees Celsius), and the associated pressure is small (tenths of a Pascal, or equivalently millionths of an atmosphere). People can hear these sounds because the cochlea is so sensitive.
One reason that microwaves might be a more plausible mechanism than sound waves for the apparent embassy attacks is acoustic impedance, discussed in Chapter 13 of IPMB. Air and water have very different impedances. When a sound wave impinges on a person, most of the acoustic energy is lost by reflection, and little (perhaps one part in a thousand) enters the fluid-filled body. Animals have evolved elaborate structures in the middle ear to mitigate this acoustic mismatch. However, a pressure wave caused by microwave heating originates inside the ear. No energy is lost by sound reflecting from the air-tissue interface.
I am no expert on thermoelastic effects, but it seems plausible that they could be responsible for the perception of sound by embassy workers in Cuba. By modifying the shape and frequency of the microwave pulses, you might even induce sounds more distinct than vague clicks. However, I don’t know how you get from little noises to brain damage and cognitive dysfunction. My brain isn’t damaged by listening to clicky sounds. Either there is more to this that I don’t understand, or—as neuroskeptic speculates—the rest of the cause is “psychological in nature.”
Right now, our country could use a hard-nosed scientist or engineer expert in the bioeffects of microwave radiation to look into this problem. Where have you gone John Moulder and Ken Foster? We need you!
I am an emeritus professor of physics at Oakland University, and coauthor of the textbook Intermediate Physics for Medicine and Biology. The purpose of this blog is specifically to support and promote my textbook, and in general to illustrate applications of physics to medicine and biology.
