Calculus 2nd Edition Rogawski

INDEX
I13
Shell Method, 323, 328
shift formulas, 30
Shifting and Scaling Rule, 151, 153
shifting:
of a graph, 8
sigma (), 247
sign:
on interval, 244
sign change, 196–197, 199, 208
sign combinations, 208
signed areas, 447
signed volume of a region, 875
simple closed curve, 878, 887
simple regions, 880
simply connected domain, 948, 949
Simpson, Thomas, 223
Simpson’s Rule, 458, 460, 520
sine function:
basic properties of, 31
derivative of, 144–145
Maclaurin expansion of, 600–601
period of, 27
unit circle definition of, 26
sine wave, 27
single integrals, 866, 874
slope field, 522–528
slope-intercept form of an equation, 13
slope of a line, 14
and polar equation, 634
smooth curve, 878
piecewise, 878
Snell’s Law, 220
solenoid:
vector potential for, 1028, 1029
solid regions:
integrating over, 894–895
solids:
cross sections of, 314
volume of, 304
solids of revolution, 314
volume of, 314–319
Solidum, Renato, 355
sound:
speed of, 44
space curve, 730, 733
space shuttle, 762
orbit of, 771
spanning with vectors, 667
Special Theory of Relativity, 403
speed, 41, 628
along parametrized path, 628
sphere:
and gradient, 825
gravitational potential of, 990
in higher dimensions, 898
and level surfaces, 788
parametrization of, 854
volume of, 365, 898–899
spherical coordinates, 721–724, 907
and earth’s magnetic field, 694, 907
level surfaces in, 720
and longitude and latitude, 721
triple integrals in, 891–896
spherical wedge, 907–908
spinors, 947
spring constant, 334
square root expressions, 426
square roots:
quadratic convergence to, 233
Squeeze Theorem, 76–79
for sequences, 543–544
stable equilibrium, 530
standard basis vectors, 669, 671, 677
standard position:
of an ellipse, 648
of a hyperbola, 649
of a parabola, 651
of quadrics, 711
steepness of the line, 13–14
Stokes’ Theorem, 1021–1030
stopping distance, 130
stream lines, see integral curves
strictly decreasing function, 6
strictly increasing function, 6
subintervals, 245
substitution:
and evaluating limits, 794–795
with hyperbolic functions, 433
and Maclaurin series, 598
and partial fractions, 438
trigonometic, 426–430
using differentials, 286–289
Substitution Method, 66–68, 286
Sum Law, 58–60, 95, 794
proof of, 95
Sum Law for Limits, 113
Sum Rule, 113–114, 236, 738–739
summation notation, 247–250
sums, partial, 554, 555, 566, 576–577
sun:
surface area, 629, 985–989
surface element, 987
surface independence:
for curl vector fields, 1027
surface integrals, 946–950, 990
over a graph, 998
of vector fields, 996–1003
surface of revolution, 480
surfaces, 1042
and boundaries, 1027
boundary of, 1021
closed, 1021
degenerate quadric, 715
flow rate of a fluid through, 1000
helicoid, 984
intersection of, 730–731
level, 720
nondegenerate quadric, 715
orientation of, 1021
oriented, 995
parametrized, 980–989
parametrizing the intersection of, 730–731
quadric, 711
of revolution, 994
smooth, 1008
and Stokes’ Theorem, 1009
total charge on, 989–990
upward-pointing, 985
see also level surfaces
symbolic differential, 1009
Symmetric Four-Noid surface, 945
symmetry, 494
and parametrization, 613
Symmetry Principle, 494, 496
tail (in the plane), 663
Tait, P. G., 819
tangent, 28
hyperbolic, 401, 402
tangent function:
derivative of, 145
integral of, 429–430
tangent line approximation (Linear
Approximation), 176
tangent lines, 40–41, 102, 740, 811
for a curve in parametric form, 618–619
defined, 102
limit of secant lines, 43, 101
and polar equation, 114
slopes of, 43, 44, 101, 619, 808, 825
vertical, 98
tangent plane, 812, 982–983
and differentiability, 814
finding an equation of, 826
at a local extremum, 839
Tangent Rule, see Midpoint Rule, 466
tangent vectors, 744, 983
derivatives, 740–742
horizontal, on the cycloid, 741-742
plotting of, 741
tangential component:
of acceleration, 765, 766
of vector field, 958, 960, 962, 963
Tartaglia, Niccolo, 222, 232
Taylor, Brook, 499
Taylor expansions, 599, 600
Taylor polynomials, 499–508, 598, 599, 601
Taylor series, 597–609
integration of, 601
multiplying, 601
shortcuts to finding, 600–602
see also Maclaurin series
Taylor’s Theorem, 505