Appendix B — Formula Reference
The "tear-out card" — every key formula from the entire two-to-three-semester sequence in one place. Formulas are grouped by topic; the home chapter is noted in parentheses so you can return to the full derivation. A constant of integration $+C$ is implied on every indefinite integral.
B.1 Limits (Ch. 3–4)
Limit Laws
Assume $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ both exist.
| Law | Statement |
|---|---|
| Sum / difference | $\lim_{x \to a}\big(f \pm g\big) = L \pm M$ |
| Constant multiple | $\lim_{x \to a} c\,f = cL$ |
| Product | $\lim_{x \to a} (fg) = LM$ |
| Quotient | $\lim_{x \to a} \dfrac{f}{g} = \dfrac{L}{M}$, provided $M \neq 0$ |
| Power | $\lim_{x \to a} \big(f(x)\big)^n = L^n$ |
| Root | $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L}$ (if $L \ge 0$ when $n$ even) |
| Composition | $\lim_{x \to a} g(f(x)) = g(L)$ if $g$ continuous at $L$ |
Squeeze theorem. If $g(x) \le f(x) \le h(x)$ near $a$ and $\lim_{x \to a} g = \lim_{x \to a} h = L$, then $\lim_{x \to a} f = L$.
Continuity. $f$ is continuous at $a$ iff $\lim_{x \to a} f(x) = f(a)$.
Special Limits
$$\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \qquad \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$$
$$\lim_{x \to 0} \frac{\tan x}{x} = 1 \qquad \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = e \qquad \lim_{x \to 0}(1 + x)^{1/x} = e$$
$$\lim_{x \to 0^+} x \ln x = 0 \qquad \lim_{x \to \infty} \frac{\ln x}{x^p} = 0 \ (p > 0) \qquad \lim_{x \to \infty} \frac{x^n}{e^x} = 0$$
$$\lim_{x \to 0} \frac{e^x - 1}{x} = 1 \qquad \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$$
L'Hôpital's Rule (Ch. 9)
If $\lim \dfrac{f}{g}$ has the indeterminate form $\dfrac{0}{0}$ or $\dfrac{\pm\infty}{\pm\infty}$, then
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$
provided the right-hand limit exists (or is $\pm\infty$).
| Indeterminate form | Strategy |
|---|---|
| $0/0$, $\infty/\infty$ | Apply L'Hôpital directly |
| $0 \cdot \infty$ | Rewrite as $\dfrac{0}{1/\infty}$ or $\dfrac{\infty}{1/0}$ to reach $0/0$ or $\infty/\infty$ |
| $\infty - \infty$ | Combine into a single fraction (common denominator) |
| $1^\infty,\ 0^0,\ \infty^0$ | Set $y =$ expression, take $\ln$, find $\lim \ln y$, then exponentiate |
B.2 Derivatives (Ch. 6–8)
Definition
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{z \to x} \frac{f(z) - f(x)}{z - x}$$
Differentiation Rules
| Rule | Formula |
|---|---|
| Constant | $\dfrac{d}{dx}(c) = 0$ |
| Power | $\dfrac{d}{dx}(x^n) = n x^{n-1}$ (any real $n$) |
| Constant multiple | $(cf)' = c f'$ |
| Sum / difference | $(f \pm g)' = f' \pm g'$ |
| Product | $(fg)' = f'g + fg'$ |
| Quotient | $\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$ |
| Chain | $\dfrac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)$ |
| Reciprocal | $\left(\dfrac{1}{g}\right)' = -\dfrac{g'}{g^2}$ |
| Inverse function | $\big(f^{-1}\big)'(x) = \dfrac{1}{f'\!\big(f^{-1}(x)\big)}$ |
Table of Derivatives
| $f(x)$ | $f'(x)$ |
|---|---|
| $c$ | $0$ |
| $x^n$ | $n x^{n-1}$ |
| $e^x$ | $e^x$ |
| $b^x$ | $b^x \ln b$ |
| $\ln x$ | $1/x$ |
| $\ln|x|$ | $1/x$ |
| $\log_b x$ | $1/(x \ln b)$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\cot x$ | $-\csc^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
| $\arcsin x$ | $1/\sqrt{1 - x^2}$ |
| $\arccos x$ | $-1/\sqrt{1 - x^2}$ |
| $\arctan x$ | $1/(1 + x^2)$ |
| $\operatorname{arccot} x$ | $-1/(1 + x^2)$ |
| $\operatorname{arcsec} x$ | $1/\big(|x|\sqrt{x^2 - 1}\big)$ |
| $\operatorname{arccsc} x$ | $-1/\big(|x|\sqrt{x^2 - 1}\big)$ |
| $\sinh x$ | $\cosh x$ |
| $\cosh x$ | $\sinh x$ |
| $\tanh x$ | $\operatorname{sech}^2 x$ |
| $\coth x$ | $-\operatorname{csch}^2 x$ |
| $\operatorname{sech} x$ | $-\operatorname{sech} x \tanh x$ |
| $\operatorname{csch} x$ | $-\operatorname{csch} x \coth x$ |
Implicit & Logarithmic Differentiation (Ch. 8)
Implicit. Differentiate both sides of $F(x, y) = 0$ with respect to $x$, treating $y = y(x)$ and applying the chain rule (every $y$ produces a factor $\tfrac{dy}{dx}$), then solve for $\tfrac{dy}{dx}$.
Logarithmic. For $y = f(x)^{g(x)}$ or messy products/quotients: take $\ln$ of both sides, differentiate implicitly, then multiply by $y$:
$$\frac{d}{dx}\big[f^g\big] = f^g\left(g' \ln f + \frac{g f'}{f}\right)$$
B.3 Integrals (Ch. 12–17)
Table of Basic Antiderivatives
| Integral | Result |
|---|---|
| $\displaystyle\int x^n \, dx \ (n \neq -1)$ | $\dfrac{x^{n+1}}{n+1}$ |
| $\displaystyle\int \frac{1}{x}\, dx$ | $\ln|x|$ |
| $\displaystyle\int e^x \, dx$ | $e^x$ |
| $\displaystyle\int b^x \, dx$ | $\dfrac{b^x}{\ln b}$ |
| $\displaystyle\int \ln x \, dx$ | $x \ln x - x$ |
| $\displaystyle\int \sin x \, dx$ | $-\cos x$ |
| $\displaystyle\int \cos x \, dx$ | $\sin x$ |
| $\displaystyle\int \sec^2 x \, dx$ | $\tan x$ |
| $\displaystyle\int \csc^2 x \, dx$ | $-\cot x$ |
| $\displaystyle\int \sec x \tan x \, dx$ | $\sec x$ |
| $\displaystyle\int \csc x \cot x \, dx$ | $-\csc x$ |
| $\displaystyle\int \tan x \, dx$ | $\ln|\sec x| = -\ln|\cos x|$ |
| $\displaystyle\int \cot x \, dx$ | $\ln|\sin x|$ |
| $\displaystyle\int \sec x \, dx$ | $\ln|\sec x + \tan x|$ |
| $\displaystyle\int \csc x \, dx$ | $-\ln|\csc x + \cot x|$ |
| $\displaystyle\int \sinh x \, dx$ | $\cosh x$ |
| $\displaystyle\int \cosh x \, dx$ | $\sinh x$ |
| $\displaystyle\int \frac{dx}{1 + x^2}$ | $\arctan x$ |
| $\displaystyle\int \frac{dx}{\sqrt{1 - x^2}}$ | $\arcsin x$ |
| $\displaystyle\int \frac{dx}{|x|\sqrt{x^2 - 1}}$ | $\operatorname{arcsec}|x|$ |
Standard Forms (with constant $a > 0$)
$$\int \frac{du}{a^2 + u^2} = \frac{1}{a}\arctan\frac{u}{a} + C$$
$$\int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin\frac{u}{a} + C$$
$$\int \frac{du}{u\sqrt{u^2 - a^2}} = \frac{1}{a}\operatorname{arcsec}\frac{|u|}{a} + C$$
$$\int \frac{du}{a^2 - u^2} = \frac{1}{2a}\ln\left|\frac{u + a}{u - a}\right| + C \qquad \int \frac{du}{u^2 - a^2} = \frac{1}{2a}\ln\left|\frac{u - a}{u + a}\right| + C$$
$$\int \frac{du}{\sqrt{u^2 \pm a^2}} = \ln\left|u + \sqrt{u^2 \pm a^2}\right| + C$$
Integration by Parts (Ch. 15)
$$\int u \, dv = uv - \int v \, du \qquad \int_a^b u \, dv = \big[uv\big]_a^b - \int_a^b v \, du$$
Choose $u$ by LIATE (Logarithmic, Inverse-trig, Algebraic, Trigonometric, Exponential — pick $u$ earliest in the list).
Common Trigonometric Integrals (Ch. 16)
Power-reduction (half-angle) identities:
$$\sin^2 x = \frac{1 - \cos 2x}{2} \qquad \cos^2 x = \frac{1 + \cos 2x}{2}$$
| Integral | Approach |
|---|---|
| $\int \sin^m x \cos^n x\, dx$, $m$ or $n$ odd | Peel one factor, convert rest via $\sin^2 + \cos^2 = 1$, substitute |
| $\int \sin^m x \cos^n x\, dx$, both even | Apply power-reduction identities |
| $\int \tan^m x \sec^n x\, dx$, $n$ even | Save $\sec^2 x$, convert with $\sec^2 = 1 + \tan^2$, let $u = \tan x$ |
| $\int \tan^m x \sec^n x\, dx$, $m$ odd | Save $\sec x \tan x$, convert with $\tan^2 = \sec^2 - 1$, let $u = \sec x$ |
Trigonometric Substitution (Ch. 16)
| Radical in integrand | Substitution | Identity used |
|---|---|---|
| $\sqrt{a^2 - u^2}$ | $u = a\sin\theta$ | $1 - \sin^2\theta = \cos^2\theta$ |
| $\sqrt{a^2 + u^2}$ | $u = a\tan\theta$ | $1 + \tan^2\theta = \sec^2\theta$ |
| $\sqrt{u^2 - a^2}$ | $u = a\sec\theta$ | $\sec^2\theta - 1 = \tan^2\theta$ |
Partial Fractions (Ch. 16)
For a proper rational function $\tfrac{P(x)}{Q(x)}$ (degree $P <$ degree $Q$; otherwise divide first), decompose by factors of $Q$:
| Factor of $Q(x)$ | Term(s) in decomposition |
|---|---|
| Distinct linear $(x - r)$ | $\dfrac{A}{x - r}$ |
| Repeated linear $(x - r)^k$ | $\dfrac{A_1}{x - r} + \dfrac{A_2}{(x - r)^2} + \cdots + \dfrac{A_k}{(x - r)^k}$ |
| Distinct irreducible quadratic $(x^2 + bx + c)$ | $\dfrac{Bx + C}{x^2 + bx + c}$ |
| Repeated irreducible quadratic, power $k$ | $\displaystyle\sum_{j=1}^{k} \dfrac{B_j x + C_j}{(x^2 + bx + c)^{j}}$ |
Fundamental Theorem of Calculus (Ch. 14)
Part 1. If $g(x) = \displaystyle\int_a^x f(t)\, dt$ with $f$ continuous, then $g'(x) = f(x)$.
Part 2. If $F' = f$, then $\displaystyle\int_a^b f(x)\, dx = F(b) - F(a)$.
Leibniz rule (variable limits). $\dfrac{d}{dx}\displaystyle\int_{u(x)}^{v(x)} f(t)\, dt = f(v(x))\,v'(x) - f(u(x))\,u'(x)$.
B.4 Applications of Integration (Ch. 18)
| Quantity | Formula |
|---|---|
| Area between curves | $\displaystyle\int_a^b \big[f(x) - g(x)\big]\, dx$ (top minus bottom) |
| Volume — disk | $\displaystyle V = \pi \int_a^b \big[R(x)\big]^2 \, dx$ |
| Volume — washer | $\displaystyle V = \pi \int_a^b \Big(\big[R(x)\big]^2 - \big[r(x)\big]^2\Big)\, dx$ |
| Volume — cylindrical shell | $\displaystyle V = 2\pi \int_a^b x\, f(x)\, dx$ (about $y$-axis) |
| Arc length ($y = f(x)$) | $\displaystyle L = \int_a^b \sqrt{1 + \big[f'(x)\big]^2}\, dx$ |
| Surface area (revolve about $x$-axis) | $\displaystyle S = 2\pi \int_a^b f(x)\sqrt{1 + \big[f'(x)\big]^2}\, dx$ |
| Surface area (revolve about $y$-axis) | $\displaystyle S = 2\pi \int_a^b x\sqrt{1 + \big[f'(x)\big]^2}\, dx$ |
| Work (variable force) | $\displaystyle W = \int_a^b F(x)\, dx$ |
| Work (Hooke's law spring) | $\displaystyle W = \int_0^d kx\, dx = \tfrac{1}{2}kd^2$ |
| Average value of $f$ on $[a,b]$ | $\displaystyle f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\, dx$ |
B.5 Sequences & Series (Ch. 20–24)
Geometric Series
$$\sum_{n=0}^{N-1} ar^n = a\,\frac{1 - r^N}{1 - r} \qquad\quad \sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r}\ (|r| < 1)$$
$p$-series. $\displaystyle\sum_{n=1}^\infty \frac{1}{n^p}$ converges iff $p > 1$.
Telescoping. $\displaystyle\sum_{n=1}^\infty (b_n - b_{n+1}) = b_1 - \lim_{n\to\infty} b_n$.
Convergence Tests at a Glance
| Test | Use when… | Converges if | Diverges if |
|---|---|---|---|
| $n$-th term (divergence) | Always check first | — (inconclusive) | $\lim a_n \neq 0$ |
| Geometric | $a_n = ar^n$ | $|r| < 1$ | $|r| \ge 1$ |
| $p$-series | $a_n = 1/n^p$ | $p > 1$ | $p \le 1$ |
| Integral | $a_n = f(n)$, $f > 0$ decreasing | $\int_1^\infty f\, dx$ converges | $\int_1^\infty f\, dx$ diverges |
| Comparison | $0 \le a_n \le b_n$ | $\sum b_n$ converges | $\sum b_n$ diverges (with $a_n \ge b_n$) |
| Limit comparison | $\lim \tfrac{a_n}{b_n} = L \in (0,\infty)$ | $\sum b_n$ converges | $\sum b_n$ diverges |
| Ratio | $\lim\big|\tfrac{a_{n+1}}{a_n}\big| = \rho$ | $\rho < 1$ | $\rho > 1$ (inconclusive if $\rho = 1$) |
| Root | $\lim |a_n|^{1/n} = \rho$ | $\rho < 1$ | $\rho > 1$ (inconclusive if $\rho = 1$) |
| Alternating ($\sum(-1)^n b_n$) | $b_n > 0$ decreasing | $\lim b_n = 0$ | $\lim b_n \neq 0$ |
Alternating series error bound. $|S - S_N| \le b_{N+1}$.
Absolute convergence. If $\sum |a_n|$ converges, then $\sum a_n$ converges (absolutely).
Taylor & Maclaurin Series (Ch. 23)
Taylor series of $f$ about $a$: $\displaystyle f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x - a)^n$. Maclaurin is the case $a = 0$.
| Series | Expansion | Interval of convergence |
|---|---|---|
| $e^x$ | $\displaystyle\sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \cdots$ | $(-\infty, \infty)$ |
| $\sin x$ | $\displaystyle\sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ | $(-\infty, \infty)$ |
| $\cos x$ | $\displaystyle\sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$ | $(-\infty, \infty)$ |
| $\dfrac{1}{1 - x}$ | $\displaystyle\sum_{n=0}^\infty x^n = 1 + x + x^2 + \cdots$ | $(-1, 1)$ |
| $\ln(1 + x)$ | $\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ | $(-1, 1]$ |
| $\arctan x$ | $\displaystyle\sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots$ | $[-1, 1]$ |
| $(1 + x)^k$ (binomial) | $\displaystyle\sum_{n=0}^\infty \binom{k}{n} x^n$, $\ \binom{k}{n} = \dfrac{k(k-1)\cdots(k-n+1)}{n!}$ | $(-1, 1)$ (in general) |
Taylor's remainder (Lagrange form). $R_n(x) = \dfrac{f^{(n+1)}(\xi)}{(n+1)!}(x - a)^{n+1}$ for some $\xi$ between $a$ and $x$.
B.6 Parametric & Polar (Ch. 25–26)
Parametric Curves ($x = x(t),\ y = y(t)$)
$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \qquad \frac{d^2 y}{dx^2} = \frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{dx/dt}$$
$$\text{Arc length: } L = \int_\alpha^\beta \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt$$
$$\text{Surface area (about } x\text{-axis): } S = \int_\alpha^\beta 2\pi y \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt$$
Polar Coordinates ($r = f(\theta)$)
$$x = r\cos\theta, \quad y = r\sin\theta, \quad r^2 = x^2 + y^2, \quad \tan\theta = \frac{y}{x}$$
$$\frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta} \qquad \text{where } r' = \frac{dr}{d\theta}$$
$$\text{Area: } A = \frac{1}{2}\int_\alpha^\beta r^2 \, d\theta \qquad \text{Arc length: } L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\, d\theta$$
B.7 Multivariable Calculus (Ch. 29–33)
Partial Derivatives & Gradient (Ch. 29–30)
$$f_x = \frac{\partial f}{\partial x}, \qquad \nabla f = \left\langle f_x,\ f_y,\ f_z \right\rangle$$
Clairaut's theorem. If continuous, mixed partials are equal: $f_{xy} = f_{yx}$.
Directional derivative (in the direction of unit vector $\mathbf{u}$):
$$D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u} = \|\nabla f\|\cos\theta$$
The gradient $\nabla f$ points in the direction of steepest ascent; $\|\nabla f\|$ is the maximum rate of increase.
Multivariable chain rule (for $z = f(x, y)$, $x = x(t)$, $y = y(t)$):
$$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$$
Tangent plane to $z = f(x, y)$ at $(a, b)$:
$$z = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b)$$
Second-Derivative (Hessian) Test (Ch. 31)
At a critical point ($f_x = f_y = 0$), let $D = f_{xx}f_{yy} - (f_{xy})^2$:
| Condition | Conclusion |
|---|---|
| $D > 0$ and $f_{xx} > 0$ | Local minimum |
| $D > 0$ and $f_{xx} < 0$ | Local maximum |
| $D < 0$ | Saddle point |
| $D = 0$ | Inconclusive |
Multiple Integrals & Coordinate Elements (Ch. 32–33)
| System | Coordinate map | Volume / area element |
|---|---|---|
| Cartesian (2D) | — | $dA = dx\, dy$ |
| Polar | $x = r\cos\theta,\ y = r\sin\theta$ | $dA = r\, dr\, d\theta$ |
| Cartesian (3D) | — | $dV = dx\, dy\, dz$ |
| Cylindrical | $x = r\cos\theta,\ y = r\sin\theta,\ z = z$ | $dV = r\, dr\, d\theta\, dz$ |
| Spherical | $x = \rho\sin\phi\cos\theta,\ y = \rho\sin\phi\sin\theta,\ z = \rho\cos\phi$ | $dV = \rho^2 \sin\phi\, d\rho\, d\phi\, d\theta$ |
(Spherical convention: $\phi$ = angle from positive $z$-axis, $0 \le \phi \le \pi$; $\theta$ = azimuthal angle.)
Change of variables (Jacobian). For $x = x(u, v)$, $y = y(u, v)$,
$$\iint_R f\, dx\, dy = \iint_S f\big(x(u,v), y(u,v)\big)\,\left|\frac{\partial(x, y)}{\partial(u, v)}\right| du\, dv, \qquad \frac{\partial(x, y)}{\partial(u, v)} = \begin{vmatrix} x_u & x_v \\ y_u & y_v \end{vmatrix}$$
(The polar element $r$ and spherical element $\rho^2\sin\phi$ are exactly the absolute Jacobians of those transformations.)
B.8 Vector Calculus (Ch. 34–38)
Operators
$$\nabla f = \left\langle \frac{\partial f}{\partial x},\ \frac{\partial f}{\partial y},\ \frac{\partial f}{\partial z}\right\rangle \quad\text{(gradient)}$$
$$\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \quad\text{(divergence)}$$
$$\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\[2pt] \partial_x & \partial_y & \partial_z \\[2pt] F_1 & F_2 & F_3 \end{vmatrix} = \left\langle \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z},\ \ \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x},\ \ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right\rangle \quad\text{(curl)}$$
Useful identities: $\nabla \times (\nabla f) = \mathbf{0}$ and $\nabla \cdot (\nabla \times \mathbf{F}) = 0$.
Line & Surface Integrals
$$\text{Scalar line integral: } \int_C f\, ds = \int_a^b f(\mathbf{r}(t))\,\|\mathbf{r}'(t)\|\, dt$$
$$\text{Work / vector line integral: } \int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\, dt$$
$$\text{Flux through surface: } \iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n}\, dS$$
The Big Theorems (Ch. 37–38)
| Theorem | Statement |
|---|---|
| FTC, Part 2 (Ch. 14) | $\displaystyle\int_a^b F'(x)\, dx = F(b) - F(a)$ |
| Fundamental Thm. for Line Integrals | $\displaystyle\int_C \nabla f \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$ |
| Green's Theorem | $\displaystyle\oint_C P\, dx + Q\, dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$ |
| Stokes' Theorem | $\displaystyle\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ |
| Divergence Theorem | $\displaystyle\iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F})\, dV$ |
Each generalizes the FTC: integrating a "derivative" over a region equals evaluating the original quantity on the region's boundary (Ch. 38).
B.9 Special Constants
$$e \approx 2.71828\,18285 \qquad \pi \approx 3.14159\,26536 \qquad \text{Euler's identity: } e^{i\pi} + 1 = 0$$
Print this. Tape it to your wall. Refer constantly — and remember that every formula here has a picture and a derivation behind it. The reference is the destination; the chapters are the journey.