Index

This alphabetical subject index maps concepts, theorems, methods, and named figures to the chapters where they are introduced or principally developed. Because this is a digital textbook, pointers are chapter-level; a few obvious section-level pointers are noted. Cross-references ("see" and "see also") connect synonyms and related ideas. Bold terms are primary entries; indented bullets are sub-entries.

A

absolute convergence, Ch. 22
absolute extremum (global max/min), Ch. 10, 31; see also extrema
acceleration, Ch. 6; see also second derivative
as a vector, Ch. 28
algebraic limit laws, Ch. 3
alternating series, Ch. 22
alternating series test, Ch. 22
alternating series estimation, Ch. 22
antiderivative, Ch. 12; see also indefinite integral
approximation
linear, Ch. 11
quadratic, Ch. 23
Taylor polynomial, Ch. 23
numerical (Riemann/trapezoid/Simpson), Ch. 13, 16
as the soul of calculus (theme), Ch. 1, 11, 23, 40
arc length, Ch. 18, 25, 26, 28
area
between curves, Ch. 18
in polar coordinates, Ch. 26
under a curve, Ch. 13
and the definite integral, Ch. 13
asymptote, Ch. 2, 9
horizontal / vertical, Ch. 3
average rate of change, Ch. 5; see also secant line
average value of a function, Ch. 14, 18

B

Basel problem, Ch. 24
bell curve, see normal distribution
Bézier curve, Ch. 25
binomial series, Ch. 23
bisection method, Ch. 4; see also intermediate value theorem
boundary (in generalized Stokes), Ch. 38
bounded sequence, Ch. 20

C

Cauchy, Augustin-Louis, Ch. 3, 22
Cauchy sequence, Ch. 20
center of mass / centroid, Ch. 18, 32
chain rule, Ch. 7
multivariable, Ch. 30
change of variables, Ch. 33; see also substitution
Clairaut's theorem (equality of mixed partials), Ch. 29
comparison test, Ch. 22
limit comparison test, Ch. 22
completeness of the reals, Ch. 20
complex numbers, Ch. 24; see also Euler's formula
concavity, Ch. 9
second derivative test, Ch. 9
conditional convergence, Ch. 22
conic sections, Ch. 27
ellipse, Ch. 27
hyperbola, Ch. 27
parabola, Ch. 27
polar form, Ch. 26
conservative vector field, Ch. 35; see also potential function
continuity, Ch. 4
of functions of several variables, Ch. 29
one-sided, Ch. 4
convergence
of sequences, Ch. 20
of series, Ch. 21, 22
radius / interval of, Ch. 23
convergence tests, Ch. 22
critical point, Ch. 9, 31
cross product, Ch. 28
curl, Ch. 34, 37
curvature, Ch. 28
cycloid, Ch. 25
cylindrical coordinates, Ch. 33

D

definite integral, Ch. 13; see also integral
del operator ($\nabla$), Ch. 30, 34
derivative, Ch. 5, 6, 7; see also differentiation
definition (limit of difference quotient), Ch. 6
directional, Ch. 30
higher-order, Ch. 6, 9
notation (Leibniz, Lagrange, Newton), Ch. 6
partial, Ch. 29
of vector-valued function, Ch. 28
difference quotient, Ch. 5, 6
differential, Ch. 11
differential equation, Ch. 19
first-order, Ch. 19
separable, Ch. 19
logistic, Ch. 19
SIR system, Ch. 19, 39
differential forms, Ch. 38
differentiation rules, Ch. 7
power rule, Ch. 7
product rule, Ch. 7
quotient rule, Ch. 7
chain rule, Ch. 7
directional derivative, Ch. 30
discontinuity, Ch. 4
jump, removable, infinite, Ch. 4
divergence (of a vector field), Ch. 34, 37
Divergence Theorem (Gauss), Ch. 37
divergence (of series), Ch. 21, 22
test for divergence (nth-term test), Ch. 22
domain, Ch. 2
dot product, Ch. 28
double integral, Ch. 32

E

$e$ (Euler's number), Ch. 7
eccentricity, Ch. 26, 27
ellipse, Ch. 27
epsilon-delta definition, Ch. 3; see also limit
error estimation
Taylor's remainder, Ch. 23
numerical integration, Ch. 16
alternating series, Ch. 22
Euler, Leonhard, Ch. 7, 19, 24
Euler's formula ($e^{i\theta}=\cos\theta+i\sin\theta$), Ch. 11, 24
Euler's identity ($e^{i\pi}+1=0$), Ch. 24
Euler's method, Ch. 19
even and odd functions, Ch. 2
exponential function, Ch. 2, 7
derivative of, Ch. 7
extrema (maxima and minima), Ch. 9, 10, 31
local vs. global, Ch. 9, 10
Extreme Value Theorem, Ch. 4

F

first derivative test, Ch. 9
fixed point, Ch. 20
flux, Ch. 36, 37
Fourier series, Ch. 24
Fubini's theorem, Ch. 32
function, Ch. 2
composite, Ch. 2, 7
implicit, Ch. 8
inverse, Ch. 2
of several variables, Ch. 29
piecewise, Ch. 2, 4
rational, Ch. 2, 16
vector-valued, Ch. 28
Fundamental Theorem of Calculus (FTC), Ch. 14
Part 1 / Part 2, Ch. 14
generalizations of, Ch. 38
Fundamental Theorem for Line Integrals, Ch. 35

G

gamma function, Ch. 17
Gaussian integral, Ch. 32, 33
generalized Stokes' theorem, Ch. 38
geometric series, Ch. 21
gradient ($\nabla f$), Ch. 30
gradient descent (anchor example), Ch. 6, 30
Green's Theorem, Ch. 35

H

harmonic series, Ch. 21, 22
Hessian matrix, Ch. 31
higher-order derivatives, Ch. 6, 9
hyperbola, Ch. 27
hyperbolic functions, Ch. 7

I

implicit differentiation, Ch. 8
improper integral, Ch. 17
type I (infinite limits), Ch. 17
type II (discontinuous integrand), Ch. 17
indefinite integral, Ch. 12; see also antiderivative
inflection point, Ch. 9
initial value problem, Ch. 12, 19
instantaneous rate of change, Ch. 5, 6; see also derivative
integral, Ch. 12–18
definite, Ch. 13
indefinite, Ch. 12
improper, Ch. 17
iterated / multiple, Ch. 32
line, Ch. 35
surface, Ch. 36
integral test, Ch. 22
integration by parts, Ch. 15
integration techniques, Ch. 15, 16
partial fractions, Ch. 16
trigonometric integrals, Ch. 16
trigonometric substitution, Ch. 16
u-substitution, Ch. 15
numerical, Ch. 16
interval of convergence, Ch. 23
Intermediate Value Theorem (IVT), Ch. 4
inverse function, Ch. 2
derivative of, Ch. 7

J

Jacobian, Ch. 33
jump discontinuity, Ch. 4

K

Kepler, Johannes, Ch. 27
Kepler's laws, Ch. 27, 28

L

Lagrange, Joseph-Louis, Ch. 23, 31
Lagrange multipliers, Ch. 31
Lagrange remainder (Taylor), Ch. 23
Leibniz, Gottfried Wilhelm, Ch. 1, 6, 14
Leibniz notation, Ch. 6
level curve / level set, Ch. 29
L'Hôpital's rule, Ch. 9
limit, Ch. 3
at infinity, Ch. 3
epsilon-delta definition, Ch. 3
of a function of several variables, Ch. 29
of a sequence, Ch. 20
one-sided, Ch. 3
limit comparison test, Ch. 22
line integral, Ch. 35
of a scalar field, Ch. 35
of a vector field, Ch. 35
linear approximation, Ch. 11; see also tangent line, differential
linearization, Ch. 11
logarithmic differentiation, Ch. 7
logarithmic function, Ch. 2, 7
logistic equation / growth, Ch. 19

M

Maclaurin series, Ch. 23
Maxwell, James Clerk, Ch. 34, 37
Maxwell's equations, Ch. 34, 37
Mean Value Theorem (MVT), Ch. 9
for integrals, Ch. 14
mixed partial derivatives, Ch. 29; see also Clairaut's theorem
modeling portfolio (progressive project), Ch. 39
monotonic sequence, Ch. 20
monotone convergence theorem, Ch. 20
multiple integral, Ch. 32; see also double integral, triple integral
multivariable chain rule, Ch. 30

N

natural logarithm, Ch. 2, 7
Newton, Isaac, Ch. 1, 6, 11
Newton's method, Ch. 11
Newton dot notation, Ch. 6
norm (magnitude) of a vector, Ch. 28
normal distribution (bell curve, anchor example), Ch. 13, 23
numerical integration, Ch. 16
midpoint rule, Ch. 13, 16
Simpson's rule, Ch. 16
trapezoidal rule, Ch. 16

O

one-sided limit, Ch. 3
optimization, Ch. 10, 31
constrained (Lagrange multipliers), Ch. 31
in several variables, Ch. 31
single-variable applied problems, Ch. 10
orientation (of curves/surfaces), Ch. 35, 36, 37

P

parabola, Ch. 27
parameterization
of a curve, Ch. 25, 28
of a surface, Ch. 36
parametric equations / curves, Ch. 25
calculus of (slope, arc length, area), Ch. 25
partial derivative, Ch. 29
partial fractions, Ch. 16
partial sum, Ch. 21
p-series, Ch. 22
polar coordinates, Ch. 26
area in, Ch. 26
conics in, Ch. 26
potential function, Ch. 35; see also conservative vector field
power rule, Ch. 7
power series, Ch. 23
radius / interval of convergence, Ch. 23
product rule, Ch. 7
projectile motion, Ch. 28

Q

quadric surface, Ch. 29
quotient rule, Ch. 7

R

radius of convergence, Ch. 23
range (of a function), Ch. 2
rate of change, see derivative
average, Ch. 5
instantaneous, Ch. 5, 6
ratio test, Ch. 22
rational function, Ch. 2, 16
rearrangement of series, Ch. 22
related rates, Ch. 8
removable discontinuity, Ch. 4
Riemann, Bernhard, Ch. 13
Riemann sum, Ch. 13
Rolle's theorem, Ch. 9
root test, Ch. 22

S

saddle point, Ch. 31
scalar field, Ch. 29, 34
secant line, Ch. 5; see also average rate of change
second derivative, Ch. 6, 9; see also concavity, acceleration
second derivative test (single variable), Ch. 9
second derivative test (two variables), Ch. 31
separable differential equation, Ch. 19
sequence, Ch. 20
series, Ch. 21
alternating, Ch. 22
geometric, Ch. 21
harmonic, Ch. 21
power, Ch. 23
Taylor / Maclaurin, Ch. 23
Simpson's rule, Ch. 16
SIR model (anchor example), Ch. 19, 39
slope field (direction field), Ch. 19
slope of tangent line, see derivative
spherical coordinates, Ch. 33
squeeze theorem, Ch. 3
Stokes' Theorem, Ch. 37
generalized, Ch. 38
substitution rule (u-substitution), Ch. 15; see also change of variables
surface area, Ch. 36
surface integral, Ch. 36

T

tangent line, Ch. 1, 5; see also derivative, linear approximation
tangent plane, Ch. 30
Taylor, Brook, Ch. 23
Taylor series, Ch. 23
applications of, Ch. 24
Taylor's theorem (with remainder), Ch. 23
terminal velocity, Ch. 19
trapezoidal rule, Ch. 16
trigonometric functions, Ch. 2, 7
derivatives of, Ch. 7
trigonometric integrals, Ch. 16
trigonometric substitution, Ch. 16
triple integral, Ch. 32

U

u-substitution, see substitution rule

V

vector field, Ch. 34
conservative, Ch. 35
curl of, Ch. 34, 37
divergence of, Ch. 34, 37
vector-valued function, Ch. 28
velocity, Ch. 5, 6
as a vector, Ch. 28
volume
by cross sections, Ch. 18
by cylindrical shells, Ch. 18
of revolution (disk/washer), Ch. 18
via triple integrals, Ch. 32

W

Weierstrass, Karl, Ch. 3
Weierstrass function (continuous, nowhere differentiable), Ch. 6
work, Ch. 18, 35

Z

zeros of a function, Ch. 4, 11; see also Newton's method, bisection method