Index Index
aa@\("!\), Definition
Absolute Convergence Theorem, Theorem
absolute maximum, Definition
absolute minimum, Definition
Absolute Value Theorem, Theorem
Alternating Series Test, Theorem
analytic function, Paragraph
analytic solution, Paragraph
angle of elevation, Paragraph
antiderivative, Definition
of vector-valued function, Definition
average rate of change, Definition
average value of a function, Definition
average value of function, Definition
average velocity, Paragraph
Binomial Series, Paragraph
Bisection Method, Paragraph
boundary point, Definition
bounded sequence, Definition Theorem
bounded set, Definition
carrying capacity, Example
center of mass, Paragraph Theorem Theorem Theorem Definition
chain rule
as matrix multiplication, Theorem
change of variables, Paragraph
circle of curvature, Paragraph
circulation, Paragraph
closed, Definition
closed disk, Definition
concave down, Definition
concave up, Definition
concavity, Definition Theorem Theorem Paragraph
conic sections, Paragraph
degenerate, Paragraph
ellipse, Definition
hyperbola, Definition
parabola, Definition
connected, Paragraph
simply, Paragraph
conservative field, Definition Theorem Theorem
constrained optimization, Paragraph
continuous function, Definition Theorem Item
properties, Theorem
vector-valued, Definition Theorem
contour lines, Paragraph
convergence, Item Item Item Item Theorem Theorem Definition Theorem Item Item Item Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Item Item Theorem Theorem Item Item
coordinates
cylindrical, Paragraph
polar, Paragraph
spherical, Paragraph
critical number, Definition
critical point, Definition Definition Theorem Definition
critical value
of a function of two variables, Section
cross product
and derivatives, Theorem
applications, Paragraph
applications!area of parallelogram, Paragraphs
applications!torque, Paragraphs
applications!volume of parallelepiped, Paragraphs
definition, Definition
of conservative fields, Theorem
curvature, Definition
and motion, Paragraph
equations for, Theorem
radius of, Paragraph
curve
parametrically defined, Definition
rectangular equation, Paragraph
smooth, Definition
curve sketching, Key Idea
cusp, Definition
cycloid, Example
cylinder, Definition
cylindrical coordinates, Paragraph
decreasing function, Definition Definition Key Idea
definite integral, Definition Theorem Theorem
of vector-valued function, Definition
del operator, Paragraph
dependent variable, Item
derivative, Definition Definition Definition Definition Definition Paragraph Key Idea Key Idea Key Idea Item Item Theorem Item Item Definition Definition Paragraph Theorem Theorem Item Theorem Theorem Theorem Theorem Paragraph Paragraph Theorem Paragraph Theorem Theorem Theorem Theorem Definition Item Item Item
directional, Definition Theorem Key Idea Item Definition Theorem
implicit, Theorem
mixed partial, Definition
multivariable differentiability, Item Definition
parametric equations, Key Idea
partial, Definition Definition
vector-valued functions, Definition Item Theorem
derivative!general, Definition
difference quotient, Paragraph
differentiability
functions of several variables, Definition
differentiable, Definition Item Definition
general functions, Definition
differential, Definition Key Idea
differential equation, Definition
linear, Definition
separable, Definition
direction field, Paragraph
directional derivative, Definition Theorem Key Idea Item Definition Theorem
directrix, Definition Definition
Disk Method, Key Idea
distance
between lines, Item
between point and line, Item
between point and plane, Key Idea
between points in space, Definition
traveled, Theorem
divergence, Item Item Item Item Theorem Theorem Definition Item Item Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Paragraph Definition
Divergence Theorem
in space, Theorem
in the plane, Theorem
dot product
and derivatives, Theorem
definition, Definition
double integral, Definition Theorem
in polar, Key Idea
properties, Theorem
eccentricity, Definition Definition
elementary function, Paragraph
ellipse
definition, Definition
eccentricity, Definition
parametric equations, Item
reflective property, Theorem
standard equation, Key Idea
extrema, Definition Item Key Idea Theorem Theorem
absolute, Definition
relative, Definition Theorem
extreme values, Definition
factorial, Definition
First Derivative Test, Theorem
first octant, Paragraph
floor function, Example
flow, Paragraph Definition
fluid pressure/force, Definition Key Idea
focus, Definition Definition Definition
Fubini's Theorem, Theorem
function
of three variables, Definition
of two variables, Definition
vector-valued, Definition
Gabriel's Horn, Example
Gauss's Law, Example
general solution, Definition
Generalized Power Rule, Theorem
geometric series, Definition Item
gradient, Definition Key Idea Item Definition Theorem Definition
and level curves, Item
and level surfaces, Theorem
Harmonic Series, Item
Head To Tail Rule, Paragraph
Hooke's Law, Paragraph
hyperbola
definition, Definition
eccentricity, Definition
parametric equations, Item
reflective property, Theorem
standard equation, Key Idea
image
of a point, Definition
of a subset, Definition
implicit solution, Definition
improper integration, Item Definition
incompressible vector field, Paragraph
increasing function, Definition Definition Key Idea
indefinite integral, Definition
of vector-valued function, Definition
independent variable, Item
inflection point, Definition
initial condition, Definition
initial point, Definition
initial value problem, Definition Paragraph
Integral Test, Theorem
integrating factor, Paragraph
integration, Definition Paragraph Item Item Definition Definition Definition Theorem Theorem Theorem Theorem Paragraph Paragraph Paragraph Theorem Definition Paragraph Paragraph Paragraph Paragraph Key Idea Key Idea Key Idea Theorem Theorem Theorem Theorem Theorem Theorem Key Idea Key Idea Item Key Idea Item Item Item Definition Item Theorem Key Idea Theorem Theorem Key Idea Key Idea Key Idea Key Idea Key Idea Key Idea Key Idea Key Idea Definition Key Idea Item
distance traveled, Theorem
double, Definition
iterated, Definition
multiple, Definition
notation, Definition
of multivariable functions, Paragraph
of vector-valued function, Definition
of vector-valued functions, Theorem
triple, Definition Theorem Definition Definition Theorem
with cylindrical coordinates, Theorem
with spherical coordinates, Theorem
interior point, Definition
Intermediate Value Theorem, Theorem
interval of convergence, Item
inverse
of a transformation, Paragraph
iterated integration, Definition Definition Theorem Definition Theorem Definition Definition Theorem
changing order, Paragraph
Lagrange multipliers, Paragraph
lamina, Paragraph
Left/Right Hand Rule, Key Idea
limit, Paragraph Paragraph Paragraph Paragraph Definition Theorem Theorem Paragraph Item Item Item Item Theorem Definition Paragraph Item Theorem Item Paragraph Key Idea Definition Theorem
of multivariable function, Definition Theorem Item
of vector-valued functions, Definition Item
properties, Theorem
line integral
over scalar field, Definition Item Paragraph
over vector field, Definition
path independent, Definition Theorem
properties over a scalar field, Item
properties over a vector field, Item
linear function, Definition
linearization, Paragraph
functions of several variables, Definition
lines, Paragraph
distances between, Item
equations for, Definition
intersecting, Paragraph
parallel, Paragraph
skew, Paragraph
logarithmic differentiation, Paragraph
logistic differential equation, Example
Maclaurin Polynomial, Definition
Maclaurin Polynomial|see{Taylor Polynomial}, Definition
Maclaurin Series, Definition
Maclaurin Series|see{Taylor Series}, Definition
magnitude of vector, Definition
center of, Paragraph Definition
matrix
Jacobian, Definition
maximum, Definition Item Theorem Theorem
absolute, Definition
relative/local, Definition Theorem
minimum, Definition Item Theorem Theorem
absolute, Definition
relative/local, Definition Theorem
moment, Definition Theorem Definition
monotonic sequence, Item
multi-index notation, Paragraph
multiple integration|see{iterated integration}, Definition
multivariable function, Definition Definition
differentiability, Item Theorem Theorem Definition Item
domain, Definition Definition
level curves, Paragraph
level surface, Paragraph
limit, Definition Theorem Item
range, Definition Definition
Möbius band, Paragraph
Newton's Method, Key Idea
norm, Definition
normal line, Definition Definition Definition
normal vector, Paragraph
nth@\(n^\text{ th }\)–term test, Theorem
numerical integration, Paragraph Paragraph Paragraph Paragraph Key Idea Key Idea Key Idea Theorem Theorem
numerical solution to a differential equation, Paragraph
octant
first, Paragraph
one to one, Subsection
one-to-one, Definition
onto, Definition
open, Definition
open ball, Item
open disk, Definition
optimization, Paragraph
constrained, Paragraph
with Lagrange multipliers, Paragraph
order, Item
orientable, Paragraph
orientation, Item
orthogonal, Definition Definition
decomposition, Key Idea
orthogonal decomposition of vectors, Key Idea
orthogonal projection, Definition
osculating circle, Paragraph
outer unit normal vector, Subsection
parabola
definition, Definition
general equation, Key Idea
reflective property, Item
parallel vectors, Item
Parallelogram Law, Paragraph
parametric equations
arc length, Theorem
concavity, Paragraph
definition, Definition
finding \(\frac{d^2y}{dx^2}\), Key Idea
finding \(\frac{dy}{dx}\), Key Idea
normal line, Definition
of a surface, Definition
surface area, Theorem
tangent line, Definition
parametrized surface, Definition
partial derivative, Definition Definition
high order, Paragraph
meaning, Paragraph
mixed, Definition
second derivative, Definition
total differential, Definition Definition
particular solution, Definition
path independent, Definition Theorem
perpendicular|see{orthogonal}, Definition
piecewise smooth curve, Paragraph
planes
coordinate plane, Paragraph
distance between point and plane, Key Idea
equations of, Definition
introduction, Paragraph
normal vector, Paragraph
tangent, Definition
point of inflection, Definition
polar
coordinates, Paragraph
function!arc length, Paragraph
function!gallery of graphs, Paragraph
function!surface area, Theorem
functions, Paragraph
functions!area, Paragraph
functions!area between curves, Paragraphs
functions!finding \(\frac{dy}{dx}\), Key Idea
functions!graphing, Paragraph
polar coordinates, Paragraph
plotting points, Example
potential function, Paragraph Definition
power series, Definition Theorem Item Theorem
quadric surface
definition, Definition
ellipsoid, Paragraph
elliptic cone, Paragraph
elliptic paraboloid, Paragraph
hyperbolic paraboloid, Paragraph
hyperboloid of one sheet, Paragraph
hyperboloid of two sheets, Paragraph
sphere, Paragraph
trace, Paragraph
Quotient Rule, Theorem
r@\(\mathbb{R}\), Paragraph
radius of convergence, Item
radius of curvature, Paragraph
Ratio Comparison Test, Theorem
regular value, Paragraph
Related Rates, Key Idea
related rates, Paragraph
Riemann Sum, Paragraph Paragraph Definition Theorem
right hand rule
of Cartesian coordinates, Paragraph
of the cross product, Paragraphs
Rolle's Theorem, Theorem
Root Comparison Test, Theorem
saddle point, Definition Theorem
sensitivity analysis, Paragraph
separation of variables, Paragraph
sequence, Theorem Subsection
sequences, Definition Definition Definition Definition Item Definition Theorem Item Item
series, Item Item Item Item Definition Item Item Theorem Example Paragraph Item Theorem Theorem Theorem Theorem Theorem Theorem Definition Theorem Theorem Item Item Paragraph Theorem Theorem Definition Theorem Item Item Item Definition Definition Paragraph
signed area, Definition
signed volume, Definition Theorem
simple curve, Paragraph
simply connected, Paragraph
slope field, Paragraph Definition
smooth, Definition
curve, Definition
surface, Definition
smooth curve
piecewise, Paragraph
speed, Item
sphere, Paragraph
spherical coordinates, Paragraph
Squeeze Theorem, Theorem
Stokes' Theorem, Theorem
surface, Definition
smooth, Definition
surface area, Key Idea Definition
surface integral, Definition
tangent line, Definition Definition Key Idea Definition
directional, Definition
tangent plane, Definition Definition
to a graph, Definition
Taylor polynimial
of several variables, Paragraph
Taylor Polynomial, Definition Item
Taylor Series, Definition Theorem Key Idea
Taylor's Theorem, Item
in several variables, Paragraph
terminal point, Definition
torque, Paragraphs
total differential, Definition Definition
sensitivity analysis, Paragraph
total signed area, Definition
trace, Paragraph
transformation, Paragraph Definition
triple integral, Definition Theorem Definition Definition Theorem
properties, Theorem
unbounded sequence, Definition
unbounded set, Definition
unit normal vector
an@\(a_\text{N}\), Paragraph
and curvature, Paragraph
definition, Definition
in \(\mathbb{R}^2\), Theorem
unit tangent vector
and curvature, Definition Paragraph
at@\(a_\text{T}\), Paragraph
definition, Definition
in \(\mathbb{R}^2\), Theorem
unit vector, Definition
properties, Item
standard unit vector, Item
unit normal vector, Definition
unit tangent vector, Definition
vector field, Item
conservative, Definition Theorem
curl of, Paragraph
divergence of, Paragraph Definition
over vector field, Key Idea
potential function of, Paragraph Definition
vector-valued function
algebra of, Definition
arc length, Theorem
average rate of change, Definition
continuity, Definition Theorem
definition, Definition
derivatives, Definition Item Theorem
describing motion, Item
displacement, Paragraph
distance traveled, Theorem
graphing, Paragraph
integration, Theorem
limits, Definition Item
smooth, Definition
tangent line, Definition
vectors, Definition
algebra of, Item
algebraic properties, Theorem
component form, Definition
cross product, Definition Theorem Theorem
definition, Definition
dot product, Definition Theorem Theorem
Head To Tail Rule, Paragraph
magnitude, Definition
norm, Definition
normal vector, Paragraph
orthogonal, Definition
orthogonal decomposition, Key Idea
orthogonal projection, Definition
parallel, Item
Parallelogram Law, Paragraph
resultant, Paragraph
standard unit vector, Item
unit vector, Definition Item
zero vector, Paragraph
volume, Definition Theorem Theorem
work, Key Idea Definition