Appendices Chapter 15 Important Formulas

Index

$!$ , see factorial
$\dfrac{\partial(x,y,z)}{\partial(u,v,w)}$ , see Jacobian
$\nabla$ , see del operator
Absolute Convergence Theorem Theorem 9.5.2
absolute maximum Definition 3.1.1
absolute minimum Definition 3.1.1
Absolute Value Theorem Theorem 9.1.2
acceleration Definition 12.3.1, §2.2
Alternating Harmonic Series §9.5, §9.8
Alternating Series Test
- for series Theorem 9.5.1
$a_{\mathrm{N}}$ Theorem 12.4.2, §12.5
analytic function §9.10
angle of elevation §12.3
antiderivative Definition 5.1.1
arc length Key Idea 10.1.1, Theorem 10.3.1, §10.5, Theorem 12.2.6, §12.5
arc length parameter §12.5, Theorem 12.5.1
asymptote
- horizontal Definition 1.5.4
- vertical §1.5
$a_{\mathrm{T}}$ Theorem 12.4.2, §12.5
average rate of change Definition 12.1.3
average value of a function Definition 14.2.2
average value of function Definition 5.4.1
Binomial Series §9.10
Bisection Method §1.6
boundary point Definition 13.2.1
bounded sequence Definition 9.1.4
- convergence Theorem 9.1.5
bounded set Definition 13.2.1
center of mass §14.4, Theorem 14.4.1, Theorem 14.4.2, Theorem 14.4.3, Definition 14.6.4
Chain Rule Theorem 2.5.1
- multivariable Theorem 13.5.1, Theorem 13.5.2
- notation §2.5
change of variable §14.7, Theorem 14.7.3
circle of curvature §12.5
circulation §15.4
closed Definition 13.2.1
closed disk Definition 13.2.1
concave down Definition 3.4.1
concave up Definition 3.4.1
concavity §10.3, Definition 3.4.1
- inflection point Theorem 3.4.1
- test for Theorem 3.4.1
conic sections §10.0
- degenerate §10.0
- ellipse Definition 10.0.2
- hyperbola Definition 10.0.3
- parabola Definition 10.0.1
connected §15.3
- simply §15.3
conservative field Definition 15.3.2, Theorem 15.3.2, Theorem 15.3.3
Constant Multiple Rule
- of derivatives Theorem 2.3.2
- of integration §5.1
- of series Theorem 9.2.2
constrained critical point §13.9
constrained optimization §13.8
continuity
- left Definition 1.6.2
- right Definition 1.6.2
continuous function Definition 1.6.1, Definition 13.2.3
- properties Theorem 1.6.3, Theorem 13.2.3
- vector-valued Definition 12.2.2, Theorem 12.2.2
contour lines §13.1
convergence
- absolute Definition 9.5.2, Theorem 9.5.2
- Alternating Series Test Theorem 9.5.1
- conditional Definition 9.5.2
- Direct Comparison Test Theorem 9.4.1
  - for integration Theorem 8.6.1
- Integral Test Theorem 9.3.1
- interval of Definition 9.8.2
- Limit Comparison Test Theorem 9.4.2
  - for integration Theorem 8.6.2
- of geometric series Theorem 9.2.1
- of improper int. Definition 8.6.1, Key Idea 8.6.1, Theorem 8.6.1, Theorem 8.6.2
- of monotonic sequences Theorem 9.1.6
- of $p$ -series Theorem 9.3.2
- of power series Theorem 9.8.1
- of sequence Definition 9.1.2, Theorem 9.1.5
- of series Definition 9.2.1
- radius of Definition 9.8.2
- Ratio Comparison Test Theorem 9.6.1
- Root Comparison Test Theorem 9.6.2
coordinates
- cylindrical §14.7
- ellipsoidal 23.
- spherical §14.7
critical number Definition 3.1.3
critical point Definition 13.8.2, Definition 13.8.3, Theorem 13.8.1, Definition 3.1.3
critical value Definition 3.1.3
cross product
- and derivatives Theorem 12.2.4
- applications §11.4
  - area of parallelogram §11.4
  - torque §11.4
  - volume of parallelepiped §11.4
- definition Definition 11.4.1
- properties Theorem 11.4.1, Theorem 11.4.2
curl §15.2, Definition 15.2.3
- of conservative fields Theorem 15.3.3
curvature Definition 12.5.1
- and motion §12.5
- equations for Theorem 12.5.2
- of circle §12.5, Example 12.5.3
- radius of §12.5
curve
- parametrically defined Definition 10.2.1
- rectangular equation §10.2
- smooth Definition 10.2.2
curve sketching Key Idea 3.5.1
cusp Definition 10.2.2
cycloid Example 12.1.4
cylinder Definition 11.1.2
cylindrical coordinates §14.7
decreasing function Definition 3.3.1
- finding intervals Key Idea 3.3.1
- strictly Definition 3.3.1
definite integral Definition 5.2.1
- and substitution Theorem 5.5.3
del operator §13.6, §15.2
derivative
- acceleration Key Idea 2.2.1
- as a function Definition 2.1.3
- at a point Definition 2.1.1
- basic rules Theorem 2.3.1
- Chain Rule Theorem 13.5.1, Theorem 13.5.2, §2.5, Theorem 2.5.1
- Constant Multiple Rule Theorem 2.3.2
- Constant Rule Theorem 2.3.1
- differential Definition 4.3.1
- directional Definition 13.6.1, Definition 13.6.3, Key Idea 13.6.1, Theorem 13.6.1, Theorem 13.6.2, Theorem 13.6.3
- First Deriv. Test Theorem 3.3.2
- Generalized Power Rule Theorem 2.5.2
- higher order Definition 2.3.1
  - interpretation §2.3
- hyperbolic funct. Key Idea 7.4.1
- implicit Theorem 13.5.3, §2.6
- interpretation §2.2
- inverse hyper. Key Idea 7.4.3
- inverse trig. Theorem 7.2.2
- Mean Value Theorem Theorem 3.2.1
- mixed partial Definition 13.3.2
- motion Key Idea 2.2.1
- multivariable differentiability Definition 13.4.2, Definition 13.4.4
- notation Definition 2.1.3, Definition 2.3.1
- parametric equations Key Idea 10.3.1
- partial Definition 13.3.1, Definition 13.3.3
- Power Rule Theorem 2.3.1
- power series Theorem 9.8.3
- Product Rule Theorem 2.4.1
- Quotient Rule Theorem 2.4.2
- Second Deriv. Test Theorem 3.4.3
- Sum/Difference Rule Theorem 2.3.2
- tangent line Definition 2.1.2
- trigonometric functions Theorem 2.4.3
- vector-valued functions Definition 12.2.3, Theorem 12.2.3, Theorem 12.2.4
- velocity Key Idea 2.2.1
differentiable Definition 13.4.2, Definition 13.4.4, §2.1
differential Definition 4.3.1
- notation Key Idea 4.3.1
Direct Comparison Test
- for integration Theorem 8.6.1
- for series Theorem 9.4.1
directional derivative Definition 13.6.1, Definition 13.6.3, Key Idea 13.6.1, Theorem 13.6.1, Theorem 13.6.2, Theorem 13.6.3
directrix Definition 10.0.1, Definition 11.1.2
discontinuous §1.6
Disk Method Key Idea 6.2.1
displacement §12.1, §12.2, §5.4
distance
- between lines Key Idea 11.5.1
- between point and line Key Idea 11.5.1
- between point and plane Key Idea 11.6.1
- between points in space Definition 11.1.1
- traveled Theorem 12.3.1
divergence §15.2, Definition 15.2.2
- Alternating Series Test Theorem 9.5.1
- Direct Comparison Test Theorem 9.4.1
  - for integration Theorem 8.6.1
- Integral Test Theorem 9.3.1
- Limit Comparison Test Theorem 9.4.2
  - for integration Theorem 8.6.2
- of geometric series Theorem 9.2.1
- of improper int. Definition 8.6.1, Key Idea 8.6.1, Theorem 8.6.1, Theorem 8.6.2
- of $p$ -series Theorem 9.3.2
- of sequence Definition 9.1.2
- of series Definition 9.2.1
- Ratio Comparison Test Theorem 9.6.1
- Root Comparison Test Theorem 9.6.2
Divergence Theorem
- in space Theorem 15.7.1
- in the plane Theorem 15.4.2
dot product
- and derivatives Theorem 12.2.4
- definition Definition 11.3.1
- properties Theorem 11.3.1, Theorem 11.3.2
double integral Definition 14.2.1, Theorem 14.2.2
- in polar Key Idea 14.3.1
- properties Theorem 14.2.3
elementary function §8.7
ellipse
- definition Definition 10.0.2
- standard equation Key Idea 10.0.2
ellipsoid 23.
extrema
- absolute Definition 13.8.1, Definition 3.1.1
- and First Deriv. Test Theorem 3.3.2
- and Second Deriv. Test Theorem 3.4.3
- finding Key Idea 3.1.1
- relative Definition 13.8.1, Theorem 13.8.1, Definition 3.1.2
Extreme Value Theorem Theorem 13.8.3, Theorem 3.1.1
extreme values Definition 3.1.1
factorial §9.1
First Derivative Test Theorem 3.3.2
floor function Example 1.6.2
flow §15.4, Definition 15.4.1
fluid pressure/force Definition 6.5.1, Key Idea 6.5.1
flux §15.4, Definition 15.4.1, §15.6, Definition 15.6.2
focus Definition 10.0.1, Definition 10.0.2, Definition 10.0.3
Fubini’s Theorem Theorem 14.2.2
function
- of three variables Definition 13.1.2
- of two variables Definition 13.1.1
- vector-valued Definition 12.1.1
Fundamental Theorem of Calculus Theorem 5.4.1, Theorem 5.4.2
- and Chain Rule §5.4
Fundamental Theorem of Line Integrals §15.3, Theorem 15.3.2
Gabriel’s Horn Example 10.1.6
gamma function 48.
Gauss’s Law Example 15.7.3
Generalized Power Rule Theorem 2.5.2
geometric sequence Definition 9.1.3
geometric series Definition 9.2.2, Theorem 9.2.1
gradient Definition 13.6.2, Definition 13.6.3, Key Idea 13.6.1, Theorem 13.6.2, Theorem 13.6.3
- and level curves Theorem 13.6.2
- and level surfaces §13.6
Green’s Theorem §15.4, Theorem 15.4.1
Head To Tail Rule §11.2
Hooke’s Law §6.4
hyperbola
- definition Definition 10.0.3
- standard equation Key Idea 10.0.3
hyperbolic function
- definition Definition 7.4.1
- derivatives Key Idea 7.4.1
- identities Key Idea 7.4.1
- integrals Key Idea 7.4.1
- inverse §7.4
  - derivative Key Idea 7.4.3
  - integration Key Idea 7.4.4
  - logarithmic def. Key Idea 7.4.2
implicit differentiation Theorem 13.5.3, §2.6
improper integration Definition 8.6.1, Definition 8.6.2
incompressible vector field §15.2
increasing function Definition 3.3.1
- finding intervals Key Idea 3.3.1
- strictly Definition 3.3.1
indefinite integral Definition 5.1.2
indeterminate form §1.1, §1.5, §7.5, Key Idea 7.5.1
inflection point Definition 3.4.2
initial point Definition 11.2.1
initial value problem §5.1
Integral Test Theorem 9.3.1
integration
- arc length Key Idea 10.1.1
- area item (b), §14.1, Definition 5.2.1
- area between curves Theorem 6.1.1
- average value Definition 5.4.1
- by parts Theorem 8.1.1
- by substitution Theorem 5.5.1
- definite Definition 5.2.1
  - and substitution Theorem 5.5.3
  - Riemann Sums Theorem 5.3.2
- displacement §5.4
- distance traveled Theorem 12.3.1
- double Definition 14.2.1
- fluid force Definition 6.5.1, Key Idea 6.5.1
- Fun. Thm. of Calc. Theorem 5.4.1, Theorem 5.4.2
- general application technique Key Idea 6.0.1
- hyperbolic funct. Key Idea 7.4.1
- improper Definition 8.6.1, Definition 8.6.2, Theorem 8.6.1, Theorem 8.6.2
- indefinite Definition 5.1.2
- inverse hyper. Key Idea 7.4.4
- iterated Definition 14.1.1
- Mean Value Theorem Theorem 5.4.3
- multiple Definition 14.1.1
- notation Definition 14.1.1, §5.1, Definition 5.2.1, §5.4
- numerical §8.7
  - Left Hand Rule Key Idea 8.7.1, Theorem 8.7.1
  - Left/Right Hand Rule §8.7
  - Midpoint Rule Key Idea 8.7.1, Theorem 8.7.1
  - Right Hand Rule Key Idea 8.7.1, Theorem 8.7.1
  - Simpson’s Rule §8.7, Key Idea 8.7.1, Theorem 8.7.1
  - Trapezoidal Rule §8.7, Key Idea 8.7.1, Theorem 8.7.1
- of multivariable functions §14.1
- of power series Theorem 9.8.3
- of trig. functions Theorem 5.5.2
- of trig. powers Key Idea 8.2.2
- of trig. powers Key Idea 8.2.1
- of vector-valued functions Definition 12.2.6
- Power Rule Theorem 5.1.3
- Sum/Difference Rule §5.1
- surface area Key Idea 10.1.2, Key Idea 10.3.4, Key Idea 10.5.4
- trig. subst. Key Idea 8.3.1
- triple Definition 14.6.1, Definition 14.6.2, Definition 14.6.3, Theorem 14.6.2, Theorem 14.6.4
- volume
  - cross-sectional area Theorem 6.2.1
  - Disk Method Key Idea 6.2.1
  - Shell Method Key Idea 6.3.1
  - Washer Method Key Idea 6.2.2
- with cylindrical coordinates Theorem 14.7.1
- with spherical coordinates Theorem 14.7.2
- work Key Idea 6.4.1
interior point Definition 13.2.1
Intermediate Value Theorem Theorem 1.6.4
interval of convergence Definition 9.8.2
irreducible quadratic §8.4
iterated integration Definition 14.1.1, Definition 14.2.1, Theorem 14.2.2, Definition 14.6.1, Definition 14.6.2, Definition 14.6.3, Theorem 14.6.2, Theorem 14.6.4
- changing order §14.1
- properties Theorem 14.2.3, Theorem 14.6.3
Jacobian §14.7
Lagrange multiplier §13.9
lamina §14.4
Left Hand Rule §5.3, §5.3, §8.7
Left/Right Hand Rule Key Idea 8.7.1
- error bounds Theorem 8.7.1
level curves §13.1, Theorem 13.6.2
level surface §13.1, §13.6
L’Hôpital’s Rule item (b), Theorem 7.5.2
limit
- Absolute Value Theorem Theorem 9.1.2
- at infinity Definition 1.5.3
- definition Definition 1.2.1
- difference quotient §1.1
- does not exist §1.1, Theorem 1.4.1
- indeterminate form §1.1, §1.5, §7.5, Key Idea 7.5.1
- left handed Definition 1.4.1
- of infinity Definition 1.5.1
- of multivariable function Definition 13.2.2, Definition 13.2.4, Theorem 13.2.1
- of sequence Definition 9.1.2
- of vector-valued functions Definition 12.2.1, Theorem 12.2.1
- one sided Definition 1.4.1
- properties Theorem 1.3.1, Theorem 13.2.1
- pseudo-definition §1.1
- right handed Definition 1.4.1
- Squeeze Theorem Theorem 1.3.5
Limit Comparison Test
- for integration Theorem 8.6.2
- for series Theorem 9.4.2
line integral
- Fundamental Theorem §15.3, Theorem 15.3.2
- over scalar field Definition 15.1.1, Theorem 15.1.1, §15.3
- over vector field Definition 15.3.1
- path independent Definition 15.3.2, Theorem 15.3.2
- properties over a scalar field Theorem 15.1.2
- properties over a vector field Theorem 15.3.1
linearization §4.3
lines §11.5
- distances between Key Idea 11.5.1
- equations for Definition 11.5.1
- intersecting §11.5
- parallel §11.5
- skew §11.5
logarithmic differentiation §7.3
Maclaurin Polynomial, see also Taylor Polynomial
- definition Definition 9.9.1
Maclaurin Series, see also Taylor Series
- definition Definition 9.10.1
magnitude of vector Definition 11.2.1
mass §14.4, Definition 14.4.1, Definition 14.6.4, Definition 15.1.2
- center of §14.4, Definition 15.1.2
maximum
- absolute Definition 13.8.1, Definition 3.1.1
- and First Deriv. Test Theorem 3.3.2
- and Second Deriv. Test Theorem 3.4.3
- relative/local Definition 13.8.1, Theorem 13.8.2, Definition 3.1.2
Mean Value Theorem
- of differentiation Theorem 3.2.1
- of integration Theorem 5.4.3
Midpoint Rule §5.3, §5.3, Key Idea 8.7.1
- error bounds Theorem 8.7.1
minimum
- absolute Definition 13.8.1, Definition 3.1.1
- and First Deriv. Test Theorem 3.3.2
- and First Deriv. Test Theorem 3.4.3
- relative/local Definition 13.8.1, Theorem 13.8.2, Definition 3.1.2
Möbius band §15.5
moment Definition 14.4.2, Theorem 14.4.3, Definition 14.6.4
monotonic sequence Definition 9.1.5
multiple integration, see iterated integration
multivariable function Definition 13.1.1, Definition 13.1.2
- continuity Definition 13.2.3, Definition 13.2.4, Theorem 13.2.3, Theorem 13.4.2
- differentiability Definition 13.4.2, Definition 13.4.4, Theorem 13.4.1, Theorem 13.4.2
- domain Definition 13.1.1, Definition 13.1.2
- level curves §13.1
- level surface §13.1
- limit Definition 13.2.2, Definition 13.2.4, Theorem 13.2.1
- range Definition 13.1.1, Definition 13.1.2
nabla, see del operator
Newton’s Method Key Idea 4.4.1
norm Definition 11.2.1
normal line Definition 10.3.2, Definition 13.7.2
normal vector §11.6
numerical integration §8.7
- Left Hand Rule Key Idea 8.7.1
  - error bounds Theorem 8.7.1
- Left/Right Hand Rule §8.7
- Midpoint Rule Key Idea 8.7.1
  - error bounds Theorem 8.7.1
- Right Hand Rule Key Idea 8.7.1
  - error bounds Theorem 8.7.1
- Simpson’s Rule §8.7, Key Idea 8.7.1
  - error bounds Theorem 8.7.1
- Trapezoidal Rule §8.7, Key Idea 8.7.1
  - error bounds Theorem 8.7.1
one to one §15.5, §7.1
open Definition 13.2.1
open ball Definition 13.2.4
open disk Definition 13.2.1
optimization §4.2
- constrained §13.8
orientable §15.5
orthogonal Definition 11.3.2, Definition 13.7.2
- decomposition Key Idea 11.3.1
orthogonal decomposition of vectors Key Idea 11.3.1
orthogonal projection Definition 11.3.3
osculating circle §12.5
outer unit normal vector §15.7
$p$ -series Definition 9.3.1, Theorem 9.3.2
parabola
- definition Definition 10.0.1
- general equation Key Idea 10.0.1
parallel vectors Definition 11.2.5
Parallelogram Law §11.2
parameterized surface Definition 15.5.1
parametric equations
- arc length Theorem 10.3.1
- concavity §10.3
- definition Definition 10.2.1
- finding $\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}$ Key Idea 10.3.2
- finding $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}$ Key Idea 10.3.1
- normal line Definition 10.3.2
- of a surface Definition 15.5.1
- surface area Key Idea 10.3.4
- tangent line Definition 10.3.1
partial derivative Definition 13.3.1, Definition 13.3.3
- high order §13.3
- meaning §13.3
- mixed Definition 13.3.2
- second derivative Definition 13.3.2
- total differential Definition 13.4.1, Definition 13.4.3
path independent Definition 15.3.2, Theorem 15.3.2
perpendicular, see orthogonal
piecewise smooth curve §15.1
planes
- coordinate plane §11.1
- distance between point and plane Key Idea 11.6.1
- equations of Definition 11.6.1
- introduction §11.1
- normal vector §11.6
- tangent Definition 13.7.3
point of inflection Definition 3.4.2
polar
- coordinates §10.4
- function
  - arc length §10.5
  - gallery of graphs §10.4
  - surface area Key Idea 10.5.4
- functions §10.4
  - area §10.5
  - area between curves §10.5
  - finding $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}$ Key Idea 10.5.1
  - graphing §10.4
polar coordinates §10.4
- plotting points Example 10.4.1
potential function §15.2, Definition 15.3.3
Power Rule
- differentiation Theorem 2.3.1
- integration Theorem 5.1.3
power series Definition 9.8.1
- algebra of Theorem 9.10.2
- convergence Theorem 9.8.1
- derivatives and integrals Theorem 9.8.3
Product Rule
- differentiation Theorem 2.4.1
projectile motion §12.3, Example 12.4.7
quadric surface
- definition Definition 11.1.3
- ellipsoid §11.1
- elliptic cone §11.1
- elliptic paraboloid §11.1
- gallery §11.1—§11.1
- hyperbolic paraboloid §11.1
- hyperboloid of one sheet §11.1
- hyperboloid of two sheets §11.1
- sphere §11.1
- trace §11.1
Quotient Rule Theorem 2.4.2
$\mathbb{R}^{2}$ §11.2
$\mathbb{R}^{3}$ §11.2
radius of convergence Definition 9.8.2
radius of curvature §12.5
Ratio Comparison Test
- for series Theorem 9.6.1
rearrangements of series §9.5, Theorem 9.5.2
related rates §4.1
Riemann Sum §5.3, §5.3, Definition 5.3.1
- and definite integral Theorem 5.3.2
Right Hand Rule §5.3, §5.3, §8.7
right hand rule
- of Cartesian coordinates §11.1
Rolle’s Theorem Theorem 3.2.2
Root Comparison Test
- for series Theorem 9.6.2
saddle point Definition 13.8.3, Theorem 13.8.2
sawtooth wave 55.
scalar §11.2
Second Derivative Test Theorem 13.8.2, Theorem 3.4.3
sensitivity analysis §13.4
sequence
- Absolute Value Theorem Theorem 9.1.2
- positive §9.4
sequences
- boundedness Definition 9.1.4
- convergent Definition 9.1.2, Theorem 9.1.5, Theorem 9.1.6
- definition Definition 9.1.1
- divergent Definition 9.1.2
- limit Definition 9.1.2
- limit properties Theorem 9.1.3
- monotonic Definition 9.1.5
series
- absolute convergence Definition 9.5.2
- Absolute Convergence Theorem Theorem 9.5.2
- alternating Definition 9.5.1
  - Approximation Theorem Theorem 9.5.3
- Alternating Series Test Theorem 9.5.1
- Binomial §9.10
- conditional convergence Definition 9.5.2
- convergent Definition 9.2.1
- definition Definition 9.2.1
- Direct Comparison Test Theorem 9.4.1
- divergent Definition 9.2.1
- geometric Definition 9.2.2, Theorem 9.2.1
- Integral Test Theorem 9.3.1
- interval of convergence Definition 9.8.2
- Limit Comparison Test Theorem 9.4.2
- Maclaurin Definition 9.10.1
- $p$ -series Definition 9.3.1, Theorem 9.3.2
- partial sums Definition 9.2.1
- power Definition 9.8.1, Theorem 9.8.1
  - derivatives and integrals Theorem 9.8.3
- properties Theorem 9.2.2
- radius of convergence Definition 9.8.2
- Ratio Comparison Test Theorem 9.6.1
- rearrangements §9.5, Theorem 9.5.2
- Root Comparison Test Theorem 9.6.2
- Taylor Definition 9.10.1
- telescoping §9.2, Example 9.2.3
Shell Method Key Idea 6.3.1
signed area §5.2
signed volume Definition 14.2.1, Theorem 14.2.2
simple curve §15.3
simply connected §15.3
Simpson’s Rule §8.7, Key Idea 8.7.1
- error bounds Theorem 8.7.1
smooth Definition 12.2.5
- surface Definition 15.5.1
smooth curve Definition 10.2.2
- piecewise §15.1
speed Definition 12.3.1
sphere §11.1
spherical coordinates §14.7
square wave 65.
Squeeze Theorem Theorem 1.3.5
Stokes’ Theorem Theorem 15.7.2
Sum/Difference Rule
- of derivatives Theorem 2.3.2
- of integration §5.1
- of series Theorem 9.2.2
summation
- notation §5.3
- properties Theorem 5.3.1
surface Definition 15.5.1
- smooth Definition 15.5.1
surface area Definition 14.5.1
- of parameterized surface §15.5, Theorem 15.5.1
- solid of revolution Key Idea 10.1.2, Key Idea 10.3.4, Key Idea 10.5.4
surface integral Definition 15.6.1
surface of revolution Key Idea 11.1.2, Key Idea 11.1.3
tangent half-angle substitution 53.
tangent line Definition 10.3.1, Key Idea 10.5.1, Definition 12.2.4, Definition 2.1.2
- directional Definition 13.7.1
tangent plane Definition 13.7.3
Taylor Polynomial
- definition Definition 9.9.1
- Taylor’s Theorem Theorem 9.9.1
Taylor Series
- common series Key Idea 9.10.1
- definition Definition 9.10.1
- equality with generating function Theorem 9.10.1
- multivariable §13.7
Taylor’s Theorem Theorem 9.9.1
telescoping series §9.2, Example 9.2.3
terminal point Definition 11.2.1
total differential Definition 13.4.1, Definition 13.4.3
- sensitivity analysis §13.4
total signed area §5.2
trace §11.1
Trapezoidal Rule §8.7, Key Idea 8.7.1
- error bounds Theorem 8.7.1
triangle wave 56.
triple integral Definition 14.6.1, Definition 14.6.2, Definition 14.6.3, Theorem 14.6.2, Theorem 14.6.4
- properties Theorem 14.6.3
unbounded sequence Definition 9.1.4
unbounded set Definition 13.2.1
unit normal vector
- $a_{\mathrm{N}}$ Theorem 12.4.2
- and acceleration §12.4, Theorem 12.4.2
- and curvature §12.5
- definition Definition 12.4.2
- in $\mathbb{R}^{2}$ Theorem 12.4.1
unit tangent vector
- and acceleration §12.4, Theorem 12.4.2
- and curvature §12.5, Definition 12.5.1
- $a_{\mathrm{T}}$ Theorem 12.4.2
- definition Definition 12.4.1
- in $\mathbb{R}^{2}$ Theorem 12.4.1
unit vector Definition 11.2.4
- properties Key Idea 11.2.1
- standard unit vector Definition 11.2.6
- unit normal vector Definition 12.4.2
- unit tangent vector Definition 12.4.1
vector field Definition 15.2.1
- conservative Definition 15.3.2, Theorem 15.3.2
- curl of §15.2
- divergence of §15.2, Definition 15.2.2
- over vector field Key Idea 15.3.1
- potential function of §15.2, Definition 15.3.3
vector-valued function
- arc length Theorem 12.2.6
- average rate of change Definition 12.1.3
- continuity Definition 12.2.2, Theorem 12.2.2
- definition Definition 12.1.1
- derivatives Definition 12.2.3, Theorem 12.2.3, Theorem 12.2.4
- describing motion Definition 12.3.1
- displacement §12.1
- distance traveled Theorem 12.3.1
- graphing §12.1
- integration Definition 12.2.6
- limits Definition 12.2.1, Theorem 12.2.1
- of constant length Theorem 12.2.5, §12.3, Key Idea 12.3.1, §12.4
- projectile motion §12.3
- smooth Definition 12.2.5
- tangent line Definition 12.2.4
vectors Definition 11.2.1
- algebra of Definition 11.2.3
- algebraic properties Theorem 11.2.1
- component form Definition 11.2.2
- cross product Definition 11.4.1, Theorem 11.4.1, Theorem 11.4.2
- definition Definition 11.2.1
- dot product Definition 11.3.1, Theorem 11.3.1, Theorem 11.3.2
- Head To Tail Rule §11.2
- magnitude Definition 11.2.1
- norm Definition 11.2.1
- normal vector §11.6
- orthogonal Definition 11.3.2
- orthogonal decomposition Key Idea 11.3.1
- orthogonal projection Definition 11.3.3
- parallel Definition 11.2.5
- Parallelogram Law §11.2
- resultant §11.2
- standard unit vector Definition 11.2.6
- unit vector Definition 11.2.4, Key Idea 11.2.1
- zero vector §11.2
velocity Definition 12.3.1, §2.2
volume Definition 14.2.1, Theorem 14.2.2, Theorem 14.6.1
Washer Method Key Idea 6.2.2
work Definition 11.3.4, Key Idea 6.4.1