DOCSLIB.ORG

C, 400 Ørstead, Christian, 397 Matrix:Symmetric, 160 Adjoint

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
C, 400 Ørstead, Christian, 397 Matrix:Symmetric, 160 Adjoint Index c, 400 recursion formula, 234 classical orthogonal polynomials, 215, 219 Ørstead, Christian, 397 second kind, 233 coefficient matrix, 136 matrix:symmetric, 160 zeros, 235 cofactors, 126 Bessel’s inequality, 253 commutation operation, 160 adjoint operator, 243 Bessel, Friedrich Wilhelm, 232 complete basis, 254 Ampère,André-Marie, 397 big-Oh, 34, 491, 493 completely antisymmetric symbol, 378 amplitude, 55 binomial coefficients, 30, 487 complex differentiation, 277 analog signal, 343 binomial expansion, 30, 487 complex functions, 271 analytic function, 288 Bose-Einstein integrals, 355 multivalued, 272 angular frequency, 55, 332 boundary value problem, 167 natual logarithm, 273 anticommutative, 377 box function, 336 real and imaginary parts, 272 arclength, 385 branch cut, 273 complex numbers, 267 Argand diagram, 267 branch point, 307 addition, 268 associated Laguerre polynomials, 462 Bromwich integral, 368 Cartesian form, 268 Associated Legendre Functions, 451 Bromwich, Thomas John l’Anson, 353 complex conjugate, 269 associated Legendre functions, 227 Buckingham P Theorem, 39 imaginary part, 267 atomic bomb example, 41 Buckingham, edgar, 39 modulus, 267 autonomous, 87 multiplication, 268 cakes nth root, 269 BAC-CAB rule, 382 cylindrical, 444 nth roots of unity, 270 back of the envelope calculations, 40 rectangular, 442 polar form, 268 beats, 72 capacitor, 61 quotient, 268 Beeblebrox, Zaphod, 412 Cardano, Girolamo, 267 real part, 267 Bernoulli numbers, 355 Carlo, Giulio, 105 complex plane, 267 Bernoulli, Daniel, 166, 267 Cauchy Integral Formula, 290 conic equation, 151 Bernoulli, Jakob, 267 Cauchy principal value integral, 302 conics, 151 Bernoulli, Jakob II, 267 Cauchy’s Theorem, 284 ellipse, 152 Bernoulli, Johann, 267 Cauchy, Augustin-Louis, 278 general equation, 151, 154 Bernoulli, Johann II, 267 Cauchy-Euler equations, 72 hyperbola, 155 Bernoulli, Johann III, 267 nonhomogeneous, 75 connected set, 280 Bernoulli, Nicolaus II, 267 Cauchy-Riemann equations, 278 constant coefficient equations, 57 Bernoulli, Nikolaus I, 267 center, 95 constant coefficient equations:complex Bessel function, 428 Chain Rule, 18 roots, 59 Bessel functions, 232 characteristic equation, 58, 74, 135 constant coefficient equations:repeated first kind, 233 charge density, 398 roots, 58 Fourier-Bessel series, 235 chemical kinetics, 149 constant coefficient systems, 137 generating function, 235 chili problem, 163, 164 continuity equation, 399 identities, 234 circle of convergence, 288 contour deformation, 285 orthogonality, 234 circular membrane, 424 contour integral, 303 502 mathematical physics convergence nonhomogeneous, 67 Euler angles, 121 absolute, 474 second order, 56 Euler’s Formula, 486 conditional, 474 separable, 49 Euler’s method, 80 pointwise, 475 differential equations, 356 Euler, Leonhard, 105, 166, 267 real sequences, 467 coupled, 130 even functions, 188 uniform, 476, 478 differential operator, 56 exponential of a matrix, 136 convergence in the mean, 253 Differentiation Under Integral, 24 convergence tests, 470 dimensional analysis, 39 Faraday’s law, 398 Comparison test, 471 Dirac delta function, 329 Faraday, Michael, 397 Integral test, 471 Laplace transform, 361 Feynman’s trick, 24 Limit comparison test, 471 sifting property, 330 Feynman, Richard, 15, 24, 354 M-test, 478 Dirac, Paul Adrien Maurice, 114, 329 Fibonacci, Leonardo Pisano, 466 nth term divergence test, 470 direction field, 92 field, 112 Ratio test, 472 directional derivative, 388 filtering, 343 Root test, 473 Dirichlet boundary conditions, 238 finite wave train, 339, 344 convolution Dirichlet kernel, 205 flux, 393 Fourier transform, 339 dispersion relation, 318 Fourier analysis, 180 Laplace transform, 364 displacement current, 399 Fourier coefficients, 180 convolution theorem distribution, 329 Fourier series, 180 Fourier transform, 339 divergence, 390 complex exponential, 325 Laplace transform, 363 divergenceless, 391 Maple code, 197 coordinates domain, 281 representation on [0, 2p], 181 cylindrical, 407 domain coloring, 273 Fourier transform, 326 polar, 117 dot product, 375 convolution, 334 cosine series, 194 double factorial, 224 properties, 332 Coulomb gauge, 402 double tank problem, 148 shifitng properties, 333 coupled systems, 87 drag force, 53 Fourier, Joseph, 166 Cramer’s rule, 90, 128 Dyson, Freeman, 3 Fourier-Bessel series, 232 cross product, 375 Fourier-Legendre series, 219, 228 cube roots of unity, 271 eigenfunction expansion, 249 Fredholm alternative, 254 curl, 390 eigenvalue, 132 free fall, 45 current, 61 eigenvalue problem, 132, 135 frequency, 55, 332 current density, 398 generalized, 159 function space, 212 curvilinear coordinates, 404 eigenvector, 132 functions cutoff frequency, 343 Einstein summation convention, 380 exponential, 12 electric dipole, 413 hyperbolic, 16 d’Alembert, Jean le Rond, 166 electric field, 398 logarithmic, 12 d’Alembertian, 404 electric permittivity, 401 polynomial, 12 damped harmonic motion, 96 electric potential, 401 rational, 12 de Moivre’s Formula, 486 electromagnetic waves, 373, 400 trigonometric, 13 de Moivre, Abraham, 486 ellipse, 151 functions:inverse trigonometric functions, de Vries, Gustav, 318 elliptic integral, 104, 105 15 derivatives, 11 entire function, 288 Fundamental Theorem of Calculus, 19, 46 table, 18 epidemic model, 150 fundamental theorem of calculus, 391 determinant, 125 equilibrium, 89 diagonalization, 131, 155 center, 95 Gamma function, 230, 352 difference equation, 351 degenerate node, 97 gate function, 336, 344 differential equation focus, 96 Gauß, Carl Friedrich, 469 autonomous, 49 node, 95 Gauss’ law, 398 first order, 49, 50 saddle, 94 Gaussian function, 334, 342 linear, 49, 56 source, 95 Gaussian integral, 335 index 503 Gegenbauer polynomials, 219 Pythagorean, 14 differential equations, 356 generalized function, 329 sum and difference, 14 inverse, 367 geometric series, 27, 288, 292 tangent, 14 properties, 352 Gibbs phenomenon, 198, 204 implicit solution, 50 series summation, 353 Gibbs, Josiah Willard, 204 impulse function, 344 Laplace’s equation, 433, 448 Goeppert-Mayer, Maria, 45 unit impulse, 362 Polar coordinates, 438 gradient, 388 impulse response, 344 Rectangular coordinates, 434 Gram-Schmidt Orthogonalization, 216 inductor, 61 Laplace, Pierre-Simon, 346 gravitational potential, 223 infinite dimensional, 212 Laplace,Pierre-Simon, 448 Green’s identity, 246 initial value problem, 49, 357 Laplacian, 169, 373 Green’s Theorem in the Plane, 284 inner product, 212 polar coordinates, 425 Green,George, 284 inner product space, 213 Laurent series, 293 Gregory, James, 479 integral transforms, 347 singular part, 294 group velocity, 322 integrals, 11 Laurent, Pierre Alphonse, 293 integration limits, 20 law of cosines, 375 Hamilton, William Rowan, 399 simple substitution, 19 Law of Gravitation, 413 harmonic conjugate, 279 table, 19 LC circuit, 64 harmonic functions, 278, 449 trigonometric, 26 least squares approximation, 251 harmonics, 166 integrating factor, 50 Legendre polynomials, 219, 451 meridional, 456 integration by parts, 21, 190 generating function, 223 sectoral, 456 integration by recursion, 351 leading coefficient, 222 spherical, 456 interval of convergence, 483 normalization, 228 surface, 456 inverse Fourier transform, 326 recurrence relation, 220 tesseral, 456 inverse Laplace transform, 357, 367 recursion formula, 220 zonal, 456 irrotational, 391 Rodrigues formula, 220 Heart of Gold, 412 Legendre, Adrien-Marie, 218, 230 heat equation, 170 Jacobi polynomials, 219 Leibniz’s Theorem, 474 1D, 170, 199 Jacobian determinant, 407 Leibniz, Gottfried Wilhelm, 474 Heaviside function, 229, 359 Jordan canonical form, 158 Levi-Civita symbol, 378 step function, 359 Jordan’s lemma, 304, 311 limit theorems, 467 Heaviside, Oliver, 229 Julia set, 274 line of equilibria, 99 Heisenberg, Werner, 337 linear operator, 56 Helmholtz equation, 418 Kepler, Johannes, 166, 232 matrix representation, 122 Helmholtz’s Theorem, 401 kernel, 317 linear transformation, 121 Helmholtz, Ludwig, 401 kettle drum, 424 linearity, 56, 332 Hermite polynomials, 219, 259 Kirchoff’s rules, 61, 145 logarithm Hitchhiker’s Guide, 412 Kortweg, Diederik, 318 multi-valued, 273 Holmes, Sherlock, 465 Kortweg-deVries equation, 317 principal value, 273 holomorphic function, 278, 288 Kronecker delta, 114, 214, 381 LRC circuits, 61, 145 homogeneous, 87 Kronecker, Leopold, 114 Hooke’s law, 85 Maclaurin series, 136 hydrogen atom, 459 L’Hopital’s Rule, 468 Maclaurin, Colin, 479 hyperbolic cosine, 16 Lagrange’s identity, 246 magnetic field, 398 hyperbolic function identities, 17 Lagrange, Joseph-Louis, 232 magnetic permeability, 401 hyperbolic sine, 16 Laguerre polynomials, 219, 462 magnetic potential, 401 hyperbolic tangent, 16 Laguerre, Edmond, 462 Mandelbrot set, 274 Laplace operator, 391 mass-spring system, 54 identities Laplace Transform coupled, 86 double angle, 14 transform pairs, 348 MATLAB code, 274 half angle, 14 Laplace transform, 346 matrices, 119 product, 14 convolution, 363 matrix 504 mathematical physics determinant, 125 orbit, 91 potential function, 401 Hermitian, 132 ordinary differential equation, 49 predator-prey model, 147 identity, 121 orthogonal functions, 183 principal axes, 152 inverse, 120 orthonormal, 183 principal vectors, 154 multiplication, 119 oscillations product solutions, 173 real symmetric, 132 coupled, 85 similar, 156 damped, 66 quaternions, 397 singular, 126 forced, 71 trace, 127 underdamped, 67 radius of convergence, 288, 483 transpose, 120 overdamped, 97 Rayleigh quotient, 247 matrix system, 136 RC circuits, 62 matrix:antisymmetrc, 160 p-test,
Load more

Recommended publications
  • Math 142 – Quiz 6 – Solutions 1. (A) by Direct Calculation, Lim 1+3N 2
    Math 142 – Quiz 6 – Solutions 1. (a) By direct calculation, 1 + 3n 1 + 3 0 + 3 3 lim = lim n = = − . n→∞ n→∞ 2 2 − 5n n − 5 0 − 5 5 3 So it is convergent with limit − 5 . (b) By using f(x), where an = f(n), x2 2x 2 lim = lim = lim = 0. x→∞ ex x→∞ ex x→∞ ex (Obtained using L’Hˆopital’s Rule twice). Thus the sequence is convergent with limit 0. (c) By comparison, n − 1 n + cos(n) n − 1 ≤ ≤ 1 and lim = 1 n + 1 n + 1 n→∞ n + 1 n n+cos(n) o so by the Squeeze Theorem, the sequence n+1 converges to 1. 2. (a) By the Ratio Test (or as a geometric series), 32n+2 32n 3232n 7n 9 L = lim (16 )/(16 ) = lim = . n→∞ 7n+1 7n n→∞ 32n 7 7n 7 The ratio is bigger than 1, so the series is divergent. Can also show using the Divergence Test since limn→∞ an = ∞. (b) By the integral test Z ∞ Z t 1 1 t dx = lim dx = lim ln(ln x)|2 = lim ln(ln t) − ln(ln 2) = ∞. 2 x ln x t→∞ 2 x ln x t→∞ t→∞ So the integral diverges and thus by the Integral Test, the series also diverges. (c) By comparison, (n3 − 3n + 1)/(n5 + 2n3) n5 − 3n3 + n2 1 − 3n−2 + n−3 lim = lim = lim = 1. n→∞ 1/n2 n→∞ n5 + 2n3 n→∞ 1 + 2n−2 Since P 1/n2 converges (p-series, p = 2 > 1), by the Limit Comparison Test, the series converges.
    [Show full text]
  • 3.3 Convergence Tests for Infinite Series
    3.3 Convergence Tests for Infinite Series 3.3.1 The integral test We may plot the sequence an in the Cartesian plane, with independent variable n and dependent variable a: n X The sum an can then be represented geometrically as the area of a collection of rectangles with n=1 height an and width 1. This geometric viewpoint suggests that we compare this sum to an integral. If an can be represented as a continuous function of n, for real numbers n, not just integers, and if the m X sequence an is decreasing, then an looks a bit like area under the curve a = a(n). n=1 In particular, m m+2 X Z m+1 X an > an dn > an n=1 n=1 n=2 For example, let us examine the first 10 terms of the harmonic series 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + : n 2 3 4 5 6 7 8 9 10 1 1 1 If we draw the curve y = x (or a = n ) we see that 10 11 10 X 1 Z 11 dx X 1 X 1 1 > > = − 1 + : n x n n 11 1 1 2 1 (See Figure 1, copied from Wikipedia) Z 11 dx Now = ln(11) − ln(1) = ln(11) so 1 x 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + > ln(11) n 2 3 4 5 6 7 8 9 10 1 and 1 1 1 1 1 1 1 1 1 1 1 + + + + + + + + + < ln(11) + (1 − ): 2 3 4 5 6 7 8 9 10 11 Z dx So we may bound our series, above and below, with some version of the integral : x If we allow the sum to turn into an infinite series, we turn the integral into an improper integral.
    [Show full text]
  • 1 Improper Integrals
    July 14, 2019 MAT136 { Week 6 Justin Ko 1 Improper Integrals In this section, we will introduce the notion of integrals over intervals of infinite length or integrals of functions with an infinite discontinuity. These definite integrals are called improper integrals, and are understood as the limits of the integrals we introduced in Week 1. Definition 1. We define two types of improper integrals: 1. Infinite Region: If f is continuous on [a; 1) or (−∞; b], the integral over an infinite domain is defined as the respective limit of integrals over finite intervals, Z 1 Z t Z b Z b f(x) dx = lim f(x) dx; f(x) dx = lim f(x) dx: a t!1 a −∞ t→−∞ t R 1 R a If both a f(x) dx < 1 and −∞ f(x) dx < 1, then Z 1 Z a Z 1 f(x) dx = f(x) dx + f(x) dx: −∞ −∞ a R 1 If one of the limits do not exist or is infinite, then −∞ f(x) dx diverges. 2. Infinite Discontinuity: If f is continuous on [a; b) or (a; b], the improper integral for a discon- tinuous function is defined as the respective limit of integrals over finite intervals, Z b Z t Z b Z b f(x) dx = lim f(x) dx f(x) dx = lim f(x) dx: a t!b− a a t!a+ t R c R b If f has a discontinuity at c 2 (a; b) and both a f(x) dx < 1 and c f(x) dx < 1, then Z b Z c Z b f(x) dx = f(x) dx + f(x) dx: a a c R b If one of the limits do not exist or is infinite, then a f(x) dx diverges.
    [Show full text]
  • Series: Convergence and Divergence Comparison Tests
    Series: Convergence and Divergence Here is a compilation of what we have done so far (up to the end of October) in terms of convergence and divergence. • Series that we know about: P∞ n Geometric Series: A geometric series is a series of the form n=0 ar . The series converges if |r| < 1 and 1 a1 diverges otherwise . If |r| < 1, the sum of the entire series is 1−r where a is the first term of the series and r is the common ratio. P∞ 1 2 p-Series Test: The series n=1 np converges if p1 and diverges otherwise . P∞ • Nth Term Test for Divergence: If limn→∞ an 6= 0, then the series n=1 an diverges. Note: If limn→∞ an = 0 we know nothing. It is possible that the series converges but it is possible that the series diverges. Comparison Tests: P∞ • Direct Comparison Test: If a series n=1 an has all positive terms, and all of its terms are eventually bigger than those in a series that is known to be divergent, then it is also divergent. The reverse is also true–if all the terms are eventually smaller than those of some convergent series, then the series is convergent. P P P That is, if an, bn and cn are all series with positive terms and an ≤ bn ≤ cn for all n sufficiently large, then P P if cn converges, then bn does as well P P if an diverges, then bn does as well. (This is a good test to use with rational functions.
    [Show full text]
  • Ch 11 Vibrations and Waves Simple Harmonic Motion Simple Harmonic Motion
    Ch 11 Vibrations and Waves Simple Harmonic Motion Simple Harmonic Motion A vibration (oscillation) back & forth taking the same amount of time for each cycle is periodic. Each vibration has an equilibrium position from which it is somehow disturbed by a given energy source. The disturbance produces a displacement from equilibrium. This is followed by a restoring force. Vibrations transfer energy. Recall Hooke’s Law The restoring force of a spring is proportional to the displacement, x. F = -kx. K is the proportionality constant and we choose the equilibrium position of x = 0. The minus sign reminds us the restoring force is always opposite the displacement, x. F is not constant but varies with position. Acceleration of the mass is not constant therefore. http://www.youtube.com/watch?v=eeYRkW8V7Vg&feature=pl ayer_embedded Key Terms Displacement- distance from equilibrium Amplitude- maximum displacement Cycle- one complete to and fro motion Period (T)- Time for one complete cycle (s) Frequency (f)- number of cycles per second (Hz) * period and frequency are inversely related: T = 1/f f = 1/T Energy in SHOs (Simple Harmonic Oscillators) In stretching or compressing a spring, work is required and potential energy is stored. Elastic PE is given by: PE = ½ kx2 Total mechanical energy E of the mass-spring system = sum of KE + PE E = ½ mv2 + ½ kx2 Here v is velocity of the mass at x position from equilibrium. E remains constant w/o friction. Energy Transformations As a mass oscillates on a spring, the energy changes from PE to KE while the total E remains constant.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Euler Equation and Geodesics R
    Euler Equation and Geodesics R. Herman February 2, 2018 Introduction Newton formulated the laws of motion in his 1687 volumes, col- lectively called the Philosophiae Naturalis Principia Mathematica, or simply the Principia. However, Newton’s development was geometrical and is not how we see classical dynamics presented when we first learn mechanics. The laws of mechanics are what are now considered analytical mechanics, in which classical dynamics is presented in a more elegant way. It is based upon variational principles, whose foundations began with the work of Eu- ler and Lagrange and have been refined by other now-famous figures in the eighteenth and nineteenth centuries. Euler coined the term the calculus of variations in 1756, though it is also called variational calculus. The goal is to find minima or maxima of func- tions of the form f : M ! R, where M can be a set of numbers, functions, paths, curves, surfaces, etc. Interest in extrema problems in classical mechan- ics began near the end of the seventeenth century with Newton and Leibniz. In the Principia, Newton was interested in the least resistance of a surface of revolution as it moves through a fluid. Seeking extrema at the time was not new, as the Egyptians knew that the shortest path between two points is a straight line and that a circle encloses the largest area for a given perimeter. Heron, an Alexandrian scholar, deter- mined that light travels along the shortest path. This problem was later taken up by Willibrord Snellius (1580–1626) after whom Snell’s law of refraction is named.
    [Show full text]
  • Sequences, Series and Taylor Approximation (Ma2712b, MA2730)
    Sequences, Series and Taylor Approximation (MA2712b, MA2730) Level 2 Teaching Team Current curator: Simon Shaw November 20, 2015 Contents 0 Introduction, Overview 6 1 Taylor Polynomials 10 1.1 Lecture 1: Taylor Polynomials, Definition . .. 10 1.1.1 Reminder from Level 1 about Differentiable Functions . .. 11 1.1.2 Definition of Taylor Polynomials . 11 1.2 Lectures 2 and 3: Taylor Polynomials, Examples . ... 13 x 1.2.1 Example: Compute and plot Tnf for f(x) = e ............ 13 1.2.2 Example: Find the Maclaurin polynomials of f(x) = sin x ...... 14 2 1.2.3 Find the Maclaurin polynomial T11f for f(x) = sin(x ) ....... 15 1.2.4 QuestionsforChapter6: ErrorEstimates . 15 1.3 Lecture 4 and 5: Calculus of Taylor Polynomials . .. 17 1.3.1 GeneralResults............................... 17 1.4 Lecture 6: Various Applications of Taylor Polynomials . ... 22 1.4.1 RelativeExtrema .............................. 22 1.4.2 Limits .................................... 24 1.4.3 How to Calculate Complicated Taylor Polynomials? . 26 1.5 ExerciseSheet1................................... 29 1.5.1 ExerciseSheet1a .............................. 29 1.5.2 FeedbackforSheet1a ........................... 33 2 Real Sequences 40 2.1 Lecture 7: Definitions, Limit of a Sequence . ... 40 2.1.1 DefinitionofaSequence .......................... 40 2.1.2 LimitofaSequence............................. 41 2.1.3 Graphic Representations of Sequences . .. 43 2.2 Lecture 8: Algebra of Limits, Special Sequences . ..... 44 2.2.1 InfiniteLimits................................ 44 1 2.2.2 AlgebraofLimits.............................. 44 2.2.3 Some Standard Convergent Sequences . .. 46 2.3 Lecture 9: Bounded and Monotone Sequences . ..... 48 2.3.1 BoundedSequences............................. 48 2.3.2 Convergent Sequences and Closed Bounded Intervals . .... 48 2.4 Lecture10:MonotoneSequences .
    [Show full text]
  • Calculus Online Textbook Chapter 10
    Contents CHAPTER 9 Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 348 9.2 Polar Equations and Graphs 351 9.3 Slope, Length, and Area for Polar Curves 356 9.4 Complex Numbers 360 CHAPTER 10 Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for ex, sin x, and cos x 10.5 Power Series CHAPTER 11 Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra in Three Dimensions CHAPTER 12 Motion along a Curve 12.1 The Position Vector 446 12.2 Plane Motion: Projectiles and Cycloids 453 12.3 Tangent Vector and Normal Vector 459 12.4 Polar Coordinates and Planetary Motion 464 CHAPTER 13 Partial Derivatives 13.1 Surfaces and Level Curves 472 13.2 Partial Derivatives 475 13.3 Tangent Planes and Linear Approximations 480 13.4 Directional Derivatives and Gradients 490 13.5 The Chain Rule 497 13.6 Maxima, Minima, and Saddle Points 504 13.7 Constraints and Lagrange Multipliers 514 CHAPTER Infinite Series Infinite series can be a pleasure (sometimes). They throw a beautiful light on sin x and cos x. They give famous numbers like n and e. Usually they produce totally unknown functions-which might be good. But on the painful side is the fact that an infinite series has infinitely many terms. It is not easy to know the sum of those terms.
    [Show full text]
  • Chapter 1 Chapter 2 Chapter 3
    Notes CHAPTER 1 1. Herbert Westren Turnbull, The Great Mathematicians in The World of Mathematics. James R. Newrnan, ed. New York: Sirnon & Schuster, 1956. 2. Will Durant, The Story of Philosophy. New York: Sirnon & Schuster, 1961, p. 41. 3. lbid., p. 44. 4. G. E. L. Owen, "Aristotle," Dictionary of Scientific Biography. New York: Char1es Scribner's Sons, Vol. 1, 1970, p. 250. 5. Durant, op. cit., p. 44. 6. Owen, op. cit., p. 251. 7. Durant, op. cit., p. 53. CHAPTER 2 1. Williarn H. Stahl, '' Aristarchus of Samos,'' Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 1, 1970, p. 246. 2. Jbid., p. 247. 3. G. J. Toorner, "Ptolerny," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 11, 1975, p. 187. CHAPTER 3 1. Stephen F. Mason, A History of the Sciences. New York: Abelard-Schurnan Ltd., 1962, p. 127. 2. Edward Rosen, "Nicolaus Copernicus," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 3, 1971, pp. 401-402. 3. Mason, op. cit., p. 128. 4. Rosen, op. cit., p. 403. 391 392 NOTES 5. David Pingree, "Tycho Brahe," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 2, 1970, p. 401. 6. lbid.. p. 402. 7. Jbid., pp. 402-403. 8. lbid., p. 413. 9. Owen Gingerich, "Johannes Kepler," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 7, 1970, p. 289. 10. lbid.• p. 290. 11. Mason, op. cit., p. 135. 12. Jbid .. p. 136. 13. Gingerich, op. cit., p. 305. CHAPTER 4 1.
    [Show full text]
  • Oscillations
    CHAPTER FOURTEEN OSCILLATIONS 14.1 INTRODUCTION In our daily life we come across various kinds of motions. You have already learnt about some of them, e.g., rectilinear 14.1 Introduction motion and motion of a projectile. Both these motions are 14.2 Periodic and oscillatory non-repetitive. We have also learnt about uniform circular motions motion and orbital motion of planets in the solar system. In 14.3 Simple harmonic motion these cases, the motion is repeated after a certain interval of 14.4 Simple harmonic motion time, that is, it is periodic. In your childhood, you must have and uniform circular enjoyed rocking in a cradle or swinging on a swing. Both motion these motions are repetitive in nature but different from the 14.5 Velocity and acceleration periodic motion of a planet. Here, the object moves to and fro in simple harmonic motion about a mean position. The pendulum of a wall clock executes 14.6 Force law for simple a similar motion. Examples of such periodic to and fro harmonic motion motion abound: a boat tossing up and down in a river, the 14.7 Energy in simple harmonic piston in a steam engine going back and forth, etc. Such a motion motion is termed as oscillatory motion. In this chapter we 14.8 Some systems executing study this motion. simple harmonic motion The study of oscillatory motion is basic to physics; its 14.9 Damped simple harmonic motion concepts are required for the understanding of many physical 14.10 Forced oscillations and phenomena. In musical instruments, like the sitar, the guitar resonance or the violin, we come across vibrating strings that produce pleasing sounds.
    [Show full text]
  • Exact Solution for the Nonlinear Pendulum (Solu¸C˜Aoexata Do Pˆendulon˜Aolinear)
    Revista Brasileira de Ensino de F¶³sica, v. 29, n. 4, p. 645-648, (2007) www.sb¯sica.org.br Notas e Discuss~oes Exact solution for the nonlinear pendulum (Solu»c~aoexata do p^endulon~aolinear) A. Bel¶endez1, C. Pascual, D.I. M¶endez,T. Bel¶endezand C. Neipp Departamento de F¶³sica, Ingenier¶³ade Sistemas y Teor¶³ade la Se~nal,Universidad de Alicante, Alicante, Spain Recebido em 30/7/2007; Aceito em 28/8/2007 This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is di®erent from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75¼ (135±) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ! in terms of the complete elliptic integral of the ¯rst kind. We can conclude that for amplitudes lower than 135o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude).
    [Show full text]