Calculus Essentials

Chem 3240 · Appendix A.1

Davit Potoyan

Why calculus, in one slide

Chemistry asks two questions of any changing quantity: how fast does it change, and how much accumulates

\text{rate} = -\frac{d[\mathrm{A}]}{dt} \qquad\qquad q = \int_{T_1}^{T_2} C_p\, dT

Two pictures carry the whole subject: the derivative is a slope, the integral is an area
Two tools do most of the work: the chain rule and integration by parts
Later the same moves act on waves and quantum states; sharpen them now

The derivative is a slope

f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}

Slope of the secant through two nearby points; merge them and it becomes the tangent
Rate of change: position gives velocity, concentration gives rate
Read backwards, a step recipe: f(x+h) \approx f(x) + f'(x)\,h

Live: shrink the spacing

secant pivots into the tangent; the step prediction f(x) + f'(x)\,h misses the curve by h^2

viewof hh = Inputs.range([0.05, 1.5], {step: 0.05, value: 1.0, label: "spacing h"})

{
  const f = x => x * x, x0 = 1, slope = 2 + hh, pred = f(x0) + 2 * hh;
  const xs = d3.range(-0.5, 2.81, 0.02);
  const curve = xs.map(x => ({x, y: f(x)}));
  const sec = xs.map(x => ({x, y: f(x0) + slope * (x - x0)}));
  const tan = xs.map(x => ({x, y: f(x0) + 2 * (x - x0)}));
  return Plot.plot({
    width: 620, height: 400,
    x: {domain: [-0.5, 2.8], label: "x"}, y: {domain: [-1, 7], label: "f(x)"},
    marks: [
      Plot.line(curve, {x: "x", y: "y", stroke: "#111", strokeWidth: 2.5}),
      Plot.line(sec, {x: "x", y: "y", stroke: "#C8102E", strokeWidth: 2, strokeDasharray: "6,4"}),
      Plot.line(tan, {x: "x", y: "y", stroke: "#1f3a93", strokeWidth: 2}),
      Plot.ruleX([x0 + hh], {y1: pred, y2: f(x0 + hh), stroke: "#107895", strokeWidth: 4}),
      Plot.dot([{x: x0, y: f(x0)}, {x: x0 + hh, y: f(x0 + hh)}], {x: "x", y: "y", fill: "#C8102E", r: 5}),
      Plot.dot([{x: x0 + hh, y: pred}], {x: "x", y: "y", fill: "#107895", r: 5, symbol: "square"})
    ]
  });
}

md`secant slope = **${(2 + hh).toFixed(2)}** (tangent slope 2); step prediction error = h² = **${(hh * hh).toFixed(3)}**`

Derivatives predict: the Taylor series

f(x + a) = f(x) + a\,f'(x) + \frac{a^2}{2!}\,f''(x) + \frac{a^3}{3!}\,f'''(x) + \cdots = \sum_{n=0}^{\infty} \frac{a^n}{n!}\,\frac{d^n f}{dx^n}

Two terms: the step recipe; the a^2 term corrects for the bending
Know by heart: e^x \approx 1 + x, \sin\theta \approx \theta, \cos\theta \approx 1 - \tfrac12\theta^2
Keep the x^2 term of a potential near its minimum: a real bond becomes a harmonic oscillator

Reading the first derivative

f' > 0: rising. f' < 0: falling. The size of f' is the steepness
f' = 0: horizontal tangent, a stationary point: maximum, minimum, or shelf
Finding where f' = 0 locates the most probable position, the equilibrium bond length, the lowest energy

The second derivative is bending

f'' is the derivative of the slope: how fast the slope is changing
f'' > 0: bends up like a bowl, so a stationary point is a minimum
f'' < 0: bends down like a dome, so a stationary point is a maximum
f'' = 0 with a sign change: inflection, the bending switches

Worked: the Gaussian, read by its derivatives

f' = -2x\,e^{-x^2} vanishes only at x = 0, where f'' = -2 < 0: the peak is a maximum
f'' = (4x^2 - 2)\,e^{-x^2} = 0 at x = \pm 1/\sqrt{2}: the inflection points
Remember them: when this curve returns as a vibrational state they are the classical turning points

In time: velocity and acceleration

x' is velocity, x'' is acceleration; Newton’s F = m\,x'' makes physics live in the bending
x = \cos t gives x'' = -x: the higher it climbs, the harder it bends back. That is the harmonic oscillator

Same in space: \dfrac{d^2}{dx^2}\sin kx = -k^2 \sin kx, which is why sines solve the wave equation. In quantum mechanics the curvature of \psi is its kinetic energy: wigglier means faster

Rules of differentiation

Constant multiple: (c f)' = c f'
Sum: (f \pm g)' = f' \pm g'
Power: \dfrac{d}{dx} x^n = n x^{n-1}
Product: (fg)' = f'g + fg'
Quotient: \left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}

\frac{d}{dx} f(g(x)) = \frac{df}{dg}\cdot\frac{dg}{dx}

The chain rule: nearly every function in this course is a composition, e^{-\alpha x^2}, \sin(kx), e^{ikx}
Outer derivative times inner derivative

Worked: a product with a composition inside

Differentiate f(x) = x^2\, e^{-\alpha x^2} (product rule, chain rule inside the exponential):

\frac{df}{dx} = (x^2)'\, e^{-\alpha x^2} + x^2\, \big(e^{-\alpha x^2}\big)' = 2x\, e^{-\alpha x^2} + x^2\,(-2\alpha x)\, e^{-\alpha x^2} = 2x\,(1 - \alpha x^2)\, e^{-\alpha x^2}

Product rule for the two factors, chain rule for (e^{-\alpha x^2})' = -2\alpha x\, e^{-\alpha x^2}

f' = 0 at x = 0 and x = \pm 1/\sqrt{\alpha}: a minimum flanked by two maxima
Same moves for the Gaussian e^{-ax^2}: f' = -2ax\,e^{-ax^2}, f'' = (4a^2x^2 - 2a)\,e^{-ax^2}, inflection points at x = \pm 1/\sqrt{2a}

Derivatives you should know by heart

f(x)	f'(x)	f(x)	f'(x)
x^n	n x^{n-1}	e^{ax}	a\,e^{ax}
\sin x	\cos x	\cos x	-\sin x
\ln x	1/x	\tan x	\sec^2 x
e^{ikx}	ik\,e^{ikx}	e^{-\alpha x^2}	-2\alpha x\, e^{-\alpha x^2}

The last row is the course in miniature: plane waves and Gaussians
Behind the table: \lim_{\theta\to0} \sin\theta/\theta = 1 and \lim_{h\to0}(e^h - 1)/h = 1, which is why e^x is its own derivative

Several variables: hold the rest fixed

\frac{\partial f}{\partial x} = \lim_{h \to 0}\frac{f(x+h,\,y) - f(x,\,y)}{h}

A surface has no single slope: slice it along one variable, freeze the others
Compute with the ordinary rules; the frozen variables ride along like constants
Shorthands f_x, f_{xx}, f_{xy}, and \left(\partial P/\partial V\right)_T in thermodynamics
Mixed partials commute: f_{xy} = f_{yx}

The chain rule on a sliding shape

u(x,t) = f(x - vt): the profile f moving right at speed v. Chain rule with s = x - vt:

\frac{\partial u}{\partial x} = f'(s), \qquad \frac{\partial u}{\partial t} = -v\,f'(s), \qquad \frac{\partial^2 u}{\partial x^2} = f''(s), \qquad \frac{\partial^2 u}{\partial t^2} = v^2 f''(s)

Each t derivative brings down one factor of -v

Eliminate f'': a relation holds for every sliding shape, whatever f is

\frac{\partial^2 u}{\partial x^2} = \frac{1}{v^2}\,\frac{\partial^2 u}{\partial t^2}

This is the classical wave equation, straight out of calculus. Chapter 2 gives it physics

The integral is an area

\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x, \qquad \Delta x = \frac{b-a}{n}

The fundamental theorem

\int_a^b f(x)\,dx = F(b) - F(a), \qquad F'(x) = f(x)

Differentiation and integration are inverse operations
An infinite sum becomes a single subtraction: find an antiderivative and you are done

f(x)	F(x)		f(x)	F(x)
x^n	\dfrac{x^{n+1}}{n+1}		e^{ax}	\dfrac{1}{a}e^{ax}
\sin x	-\cos x		\cos x	\sin x

Two techniques cover almost everything

Substitution, the chain rule run backwards: \int f(g(x))\,g'(x)\,dx = \int f(u)\,du

\int u\,dv = uv - \int v\,du

Integration by parts is the product rule run backwards: \int x\,e^{-x}dx, \int x \sin x\,dx, every expectation value
For sines and cosines: \sin^2 x = \tfrac12(1 - \cos 2x), \cos^2 x = \tfrac12(1 + \cos 2x)

Symmetry kills integrals on sight

Even f(-x) = f(x): \displaystyle\int_{-a}^{a} f\,dx = 2\int_0^a f\,dx
Odd f(-x) = -f(x): \displaystyle\int_{-a}^{a} f\,dx = 0
Check the symmetry before you integrate: \langle x \rangle = 0 for every symmetric state

The two integrals you will use all semester

\int_0^L \sin^2\!\left(\frac{n\pi x}{L}\right) dx = \frac{L}{2}

Power reduction, the cosine integrates to zero over whole half-periods. This is the \sqrt{2/L} in front of every particle in a box state

\int_0^\infty x^n e^{-x}\, dx = n!

Integrate by parts n times, or recognize the Gamma function. The workhorse of hydrogen atom radial integrals

Takeaway

A derivative is a slope and a step recipe, an integral is an area, and the fundamental theorem makes them inverses. A partial derivative is the same slope with the other variables frozen. The chain rule and integration by parts do most of the work; check the symmetry of the integrand before doing either.