McGraw-Hill Dictionary of Engineering Second Edition McGraw-Hill New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2003 by The McGraw-Hill Companies, Inc All rights reserved Manufactured in the United States of America Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher 0-07-141799-0 The material in this eBook also appears in the print version of this title: 0-07-141050-3 All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark Where such designations appear in this book, they have been 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comply with these terms THE WORK IS PROVIDED “AS IS” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGraw-Hill and its licensors not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom McGraw-Hill has no responsibility for the content of any information accessed through the work Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise DOI: 10.1036/0071417990 For more information about this title, click here Contents Preface v Staff vi How to Use the Dictionary .vii Fields and Their Scope ix Pronunciation Key .xi A-Z Terms 1-626 Appendix 627-643 Equivalents of commonly used units for the U.S Customary System and the metric system 629 Conversion factors for the U.S Customary System, metric system, and International System .630 Special constants 634 Electrical and magnetic units 635 Dimensional formulas of common quantities .635 Internal energy and generalized work 636 General rules of integration 637 Schematic electronic symbols .639 Copyright 2003 by The McGraw-Hill Companies, Inc Click Here for Terms of Use This page intentionally left blank Preface The McGraw-Hill Dictionary of Engineering provides a compendium of more than 18,000 terms that are central to the various branches of engineering and related fields of science The coverage in this Second Edition is focused on building construction, chemical engineering, civil engineering, control systems, design engineering, electricity and electronics, engineering acoustics, industrial engineering, mechanics and mechanical engineering, systems engineering, and thermodynamics Many new entries have been added since the previous edition with others revised as necessary Many of the terms used in engineering are often found in specialized dictionaries and glossaries; this Dictionary, however, aims to provide the user with the convenience of a single, comprehensive reference All of the definitions are drawn from the McGraw-Hill Dictionary of Scientific and Technical Terms, Sixth Edition (2003) Each definition is classified according to the field with which it is primarily associated; if it is used in more than one area, it is idenfified by the general label [ENGINEERING] The pronunciation of each term is provided along with synonyms, acronyms, and abbreviations where appropriate A guide to the use of the Dictionary appears on pages vii and viii, explaining the alphabetical organization of terms, the format of the book, cross referencing, and how synonyms, variant spellings, abbreviations, and similar information are handled The Pronunciation Key is given on page xi The Appendix provides conversion tables for commonly used scientific units as well as listings of useful mathematical, engineering, and scientific data It is the editors’ hope that the Second Edition of the McGraw-Hill Dictionary of Engineering will serve the needs of scientists, engineers, students, teachers, librarians, and writers for high-quality information, and that it will contribute to scientific literacy and communication Mark D Licker Publisher v Copyright 2003 by The McGraw-Hill Companies, Inc Click Here for Terms of Use Staff Mark D Licker, Publisher—Science Elizabeth Geller, Managing Editor Jonathan Weil, Senior Staff Editor David Blumel, Staff Editor Alyssa Rappaport, Staff Editor Charles Wagner, Digital Content Manager Renee Taylor, Editorial Assistant Roger Kasunic, Vice President—Editing, Design, and Production Joe Faulk, Editing Manager Frank Kotowski, Jr., Senior Editing Supervisor Ron Lane, Art Director Thomas G Kowalczyk, Production Manager Pamela A Pelton, Senior Production Supervisor Henry F Beechhold, Pronunciation Editor Professor Emeritus of English Former Chairman, Linguistics Program The College of New Jersey Trenton, New Jersey vi Copyright 2003 by The McGraw-Hill Companies, Inc Click Here for Terms of Use How to Use the Dictionary ALPHABETIZATION The terms in the McGraw-Hill Dictionary of Engineering, Second Edition, are alphabetized on a letter-by-letter basis; word spacing, hyphen, comma, solidus, and apostrophe in a term are ignored in the sequencing For example, an ordering of terms would be: abat-vent A block Abney level ADP air band airblasting FORMAT The basic format for a defining entry provides the term in boldface, the field is small capitals, and the single definition in lightface: term [FIELD] Definition A field may be followed by multiple definitions, each introduced by a boldface number: term [FIELD] Definition Definition Definition A term may have definitions in two or more fields: term [CIV ENG] Definition [ENG ACOUS] Definition A simple cross-reference entry appears as: term See another term A cross reference may also appear in combination with definitions: term [CIV ENG] Definition [ENG ACOUS] Definition CROSS REFERENCING A cross-reference entry directs the user to the defining entry For example, the user looking up “access flooring” finds: access flooring See raised flooring The user then turns to the “R” terms for the definition Cross references are also made from variant spellings, acronyms, abbreviations, and symbols ARL See acceptable reliability level arriswise See arrisways at See technical atmosphere ALSO KNOWN AS , etc A definition may conclude with a mention of a synonym of the term, a variant spelling, an abbreviation for the term, or other vii Copyright 2003 by The McGraw-Hill Companies, Inc Click Here for Terms of Use such information, introduced by “Also known as ,” “Also spelled ,” “Abbreviated ,” “Symbolized ,” “Derived from ” When a term has more than one definition, the positioning of any of these phrases conveys the extent of applicability For example: term [CIV ENG] Definition Also known as synonym Definition Symbolized T In the above arrangement, “Also known as ” applies only to the first definition; “Symbolized ” applies only to the second definition term [CIV ENG] Definition Definition [ENG Also known as synonym ACOUS] Definition In the above arrangement, “Also known as ” applies only to the second field term [CIV ENG] Also known as synonym [ENG ACOUS] Definition Definition Definition In the above arrangement, “Also known as ” applies to both definitions in the first field term Also known as synonym [CIV ENG] Definition Definition [ENG ACOUS] Definition In the above arrangement, “Also known as ” applies to all definitions in both fields viii inch ϭ 2.5 centimeters (25 millimeters) foot ϭ 0.3 meter (30 centimeters) yard ϭ 0.9 meter mile ϭ 1.6 kilometers 629 quart ϭ 0.95 liter ounce ϭ 28.35 grams pound ϭ 0.45 kilogram ton ϭ 907.18 kilograms ЊF ϭ (1.8 ϫ ЊC) ϩ 32 1 1 gallon ϭ 3.8 liters fluid ounce ϭ 29.6 milliliters 32 fluid ounces ϭ 946.4 milliliters acre ϭ 0.4 hectare acre ϭ 4047 square meters 1 1 1 1 inch ϭ 0.083 foot foot ϭ 0.33 yard (12 inches) yard ϭ feet (36 inches) mile ϭ 5280 feet (1760 yards) quart ϭ 0.25 gallon (32 ounces; pints) pint ϭ 0.125 gallon (16 ounces) gallon ϭ quarts (8 pints) centimeter ϭ 0.4 inch meter ϭ 3.3 feet meter ϭ 1.1 yards kilometer ϭ 0.62 mile gram ϭ 0.035 ounce kilogram ϭ 2.2 pounds kilogram ϭ 1.1 ϫ 10Ϫ3 ton ЊC ϭ (ЊF Ϫ 32) Ϭ 1.8 ounce ϭ 0.0625 pound pound ϭ 16 ounces ton ϭ 2000 pounds liter ϭ 1.06 quarts ϭ 0.26 gallon milliliter ϭ 0.034 fluid ounce hectare ϭ 2.47 acres square meter ϭ 0.00025 acre 1 1 Equivalents of commonly used units for the U.S Customary System and the metric system Appendix 0.01 0.9144 ϭ 30.48 ϭ 91.44 ft yd 630 1296 4.014490 ϫ 109 6.4516 ϫ 10Ϫ4 0.09290304 0.8361273 2.589988 ϫ 106 ϭ 6.4516 ϭ 929.0304 ϭ 8361.273 ϭ 2.589988 ϫ 1010 in.2 ft2 yd2 mi2 144 1550.003 ϭ 104 m2 0.1550003 in.2 10Ϫ4 m2 ϭ1 cm2 6.336 ϫ 104 36 12 39.37008 0.3937008 in cm2 B Units of area Units m 1.609344 ϫ 103 0.3048 ϭ 2.54 in ϭ 1.609344 ϫ 105 0.0254 ϭ 100 1m mi ϭ1 cm A Units of length Units cm ft2 yd2 mi2 3.228306 ϫ 10Ϫ7 3.0976 ϫ 106 2.78784 ϫ 107 3.587007 ϫ 10Ϫ8 2.490977 ϫ 10Ϫ10 3.861022 ϫ 10Ϫ7 3.861022 ϫ 10Ϫ11 5.681818 ϫ 10Ϫ4 1.893939 ϫ 10Ϫ4 1.578283 ϫ 10Ϫ5 6.213712 ϫ 10Ϫ4 6.213712 ϫ 10Ϫ6 mi 0.1111111 7.716049 ϫ 10Ϫ4 1.195990 1.195990 ϫ 10Ϫ4 1760 0.3333333 0.02777777 1.093613 0.01093613 yd 6.944444 ϫ 10Ϫ3 10.76391 1.076391 ϫ 10Ϫ3 5280 0.08333333 3.280840 0.03280840 ft Conversion factors for the U.S Customary System, metric system, and International System Appendix Ϫ6 Ϫ5 631 10 0.4535924 ϭ 28.34952 ϭ 453.5924 oz (avdp) lb (avdp) ton ϭ 907184.7 907.1847 1000 0.02834952 ϭ 1000 kg metric ton ϭ 108 ϭ1 1g kg 3785.412 ϭ 3.785412 ϫ 10Ϫ3 gal (U.S.) Ϫ3 946.3529 ϭ 9.463529 ϫ 10Ϫ4 qt D Units of mass Units g 28316.85 16.38706 1000 10 cm3 ϭ 2.831685 ϫ 10Ϫ2 ϭ 1.638706 ϫ 10 ϭ 10Ϫ3 ϭ 10 ϭ1 m3 ft3 in liter cm 1m C Units of volume Units Ϫ3 oz 231 57.75 1728 61.02374 0.06102374 35.27396 32000 35273.96 16 in.3 6.102374 ϫ 10 0.03527396 3.785412 0.9463529 28.31685 0.01638706 10 10 liter lb Ϫ3 0.1336806 0.03342014 0.0625 2.204623 2204.623 2000 Ϫ4 5.787037 ϫ 10 0.03531467 Ϫ5 3.531467 ϫ 10 2.204623 ϫ 10 Ϫ3 35.31467 ϫ 10 ft3 Ϫ3 ton 3.125 ϫ 10Ϫ5 ϫ 10Ϫ4 2.834952 ϫ 10Ϫ5 4.535924 ϫ 10Ϫ4 0.9071847 1.102311 1.102311 ϫ 10Ϫ3 10Ϫ3 1.102311 ϫ 10Ϫ6 10 metric ton 0.25 7.480520 4.329004 ϫ 10Ϫ3 0.2641721 2.641721 ϫ 10Ϫ4 264.1721 gal Ϫ6 2.992208 0.01731602 1.056688 1.056688 ϫ 10 1.056688 qt Appendix ϭ1 ϭ 1.729994 ϭ 27.67991 oz и in.Ϫ3 Ϫ3 632 106 1013250 ϭ 0.1 ϭ 105 ϭ 101325 dyn и cmϪ2 bar atm 980665 33863.88 ϭ 3386.388 ϭ 6894.757 in Hg lbf и in.Ϫ2 68947.57 1333.224 ϭ 133.3224 mmHg (torr) kgf и cm ϭ 98066.5 Ϫ2 10 Pa, N и mϪ2 ϭ dyn и cmϪ2 ϭ 0.1198264 lb и galϪ1 F Units of pressure Units Pa, N и mϪ2 ϭ 0.01601847 lb и ftϪ3 lb и in ϭ 10Ϫ3 g и LϪ1, kg и mϪ3 g и cmϪ3 g и cmϪ3 E Units of density Units 0.06894757 0.03386388 1.333224 ϫ 103 0.980665 1.01325 10Ϫ6 10Ϫ5 bar 119.8264 16.01847 27679.91 1729.994 1000 g и LϪ1, kg и mϪ3 1.019716 ϫ 10Ϫ6 1.019716 ϫ 10Ϫ5 1.033227 0.06804596 0.03342105 0.07030696 0.03453155 1.359510 ϫ 10Ϫ3 1.315789 ϫ 10Ϫ3 7.500617 ϫ 10Ϫ4 51.71493 25.4 735.5592 760 in Hg 2.036021 0.03937008 28.95903 29.92126 29.52999 2.952999 ϫ 10Ϫ5 2.952999 ϫ 10Ϫ4 7.480519 1728 108 0.06242795 7.500617 ϫ 10Ϫ3 750.0617 lb и ftϪ3 62.42795 mmHg (torr) 4.329004 ϫ 10Ϫ3 5.787037 ϫ 10Ϫ4 0.0625 3.612728 ϫ 10Ϫ5 kgf и cmϪ2 1.019716 lb и in.Ϫ3 0.03612728 0.9678411 0.9869233 9.869233 ϫ 10Ϫ7 9.869233 ϫ 10Ϫ6 atm 4.749536 ϫ 10Ϫ3 9.259259 ϫ 10Ϫ3 16 5.780365 ϫ 10Ϫ4 0.5780365 oz и in.Ϫ3 Conversion factors for the U.S Customary System, metric system, and International System (cont.) 1 0.4911541 0.01933678 14.22334 14.69595 14.50377 1.450377 ϫ 10Ϫ5 1.450377 ϫ 10Ϫ4 lbf и in.Ϫ2 0.1336806 231 14.4375 8.345403 ϫ 10Ϫ3 8.345403 lb и galϪ1 Appendix 633 4.1868 1055.056 3600000 2384519 1.355818 ϭ 4.658443 ϫ 10Ϫ14 ϭ 1.173908 ϫ 10Ϫ11 ϭ 4.005540 ϫ 10Ϫ8 ϭ 2.986931 ϫ 10Ϫ8 ϭ 1.508551 ϫ 10Ϫ14 ϭ 2.172313 ϫ 10Ϫ12 calIT BtuIT kWh hp-h ft-lbf ft3 lbf и in.Ϫ2 liter-atm ϭ 1.127393 ϫ 10Ϫ12 4.184 ϭ 4.655328 ϫ 10Ϫ14 cal 101.325 6.324210 ϫ 1020 1.218579 ϫ 1021 8.462351 ϫ 1018 1.675545 ϫ 1025 2.246944 ϫ 1025 6.585141 ϫ 1021 2.613195 ϫ 1019 2.611448 ϫ 1019 1.602176 ϫ 10Ϫ19 ϭ 1.782662 ϫ 10Ϫ33 eV 6.241510 ϫ 1018 195.2378 eV 5.609589 ϫ 1032 ϭ 1.112650 ϫ 10Ϫ14 8.987552 ϫ 1013 J 1J g mass ϭ (energy equiv) G Units of energy g mass Units (energy equiv) 2.146640 ϫ 1013 calIT 641186.5 859845.2 251.9958 0.9993312 24.21726 46.66295 24.20106 46.63174 0.3240483 0.3238315 641615.6 860420.7 252.1644 1.000669 3.829293 3.826733 ϫ 10Ϫ20 ϫ 10Ϫ20 0.2390057 0.2388459 2.148076 ϫ 103 cal 0.09603757 0.1850497 1.285067 ϫ 10Ϫ3 2544.33 3412.142 3.968321 ϫ 10Ϫ3 3.965667 ϫ 10Ϫ3 1.518570 ϫ 10Ϫ22 9.478172 ϫ 10Ϫ4 8.518555 ϫ 1010 BtuIT 2.814583 ϫ 10Ϫ5 5.423272 ϫ 10Ϫ5 3.766161 ϫ 10Ϫ7 0.7456998 2.930711 ϫ 10Ϫ4 1.163 ϫ 10Ϫ6 1.1622222 ϫ 10Ϫ6 4.450490 ϫ 10Ϫ26 2.777777 ϫ 10Ϫ7 2.496542 ϫ 107 kWh 3.774419 ϫ 10Ϫ5 7.272727 ϫ 10Ϫ5 5.050505 ϫ 10Ϫ7 1.341022 3.930148 ϫ 10Ϫ4 1.559609 ϫ 10Ϫ6 1.558562 ϫ 10Ϫ6 5.968206 ϫ 10Ϫ26 3.725062 3.347918 ϫ 107 hp-h 4.603388 ϫ 1011 74.73349 144 1980000 2655224 778.1693 3.088025 3.085960 0.5189825 6.944444 ϫ 10Ϫ3 13750 18349.06 5.403953 2.144462 ϫ 10Ϫ2 2.143028 ϫ 10Ϫ2 1.181705 8.206283 ϫ 10Ϫ19 ϫ 10Ϫ22 liter-atm 1.926847 0.01338088 26494.15 35529.24 10.41259 0.04132050 0.04129287 1.581225 ϫ 10Ϫ21 9.869233 ϫ 10Ϫ3 8.870024 ϫ 1011 ft3 и lbf и in.Ϫ2 0.7375622 5.121960 ϫ 10Ϫ3 6.628878 ϫ 1013 ft-lbf Appendix Appendix Special constants ϭ 3.14159 26535 89793 23846 2643 e ϭ 2.71828 18284 59045 23536 0287 ϭ lim ϩ n→ϱ ϭ natural base of logarithms n n Ί2 ϭ 1.41421 35623 73095 0488 Ί3 ϭ 1.73205 08075 68877 2935 Ί5 ϭ 2.23606 79774 99789 6964 Ί2 ϭ 1.25992 1050 Ί3 ϭ 1.44224 9570 Ί2 ϭ 1.14869 8355 Ί3 ϭ 1.24573 0940 e ϭ 23.14069 26327 79269 006 e ϭ 22.45915 77183 61045 47342 715 e e ϭ 15.15426 22414 79264 190 log10 ϭ 0.30102 99956 63981 19521 37389 log10 ϭ 0.47712 12547 19662 43729 50279 log10 e ϭ 0.43429 44819 03251 82765 log10 ϭ 0.49714 98726 94133 85435 12683 loge 10 ϭ ln 10 ϭ 2.30258 50929 94045 68401 7991 loge ϭ ln ϭ 0.69314 71805 59945 30941 7232 loge ϭ ln ϭ 1.09861 22886 68109 69139 5245 ␥ ϭ 0.57721 56649 01532 86060 6512 ϭ Euler’s constant ϭ lim ϩ n→ϱ 1 ϩ ϩ иии ϩ Ϫ ln n n e␥ ϭ 1.78107 24179 90197 9852 Ίe ϭ 1.64872 12707 00128 1468 Ί ϭ ⌫(–) ϭ 1.77245 38509 05516 02729 8167 where ⌫ is the gamma function ⌫(–) ϭ 2.67893 85347 07748 ⌫(–) ϭ 3.62560 99082 21908 radian ϭ 180Њ/ ϭ 57.29577 95130 8232 Њ 1Њ ϭ /180 radians ϭ 0.01745 32925 19943 29576 92 radians SOURCE: Murray R Spiegel and John Liu, Mathematical Handbook of Formulas and Tables, 2d ed., Schaum’s Outline Series, McGraw-Hill, 1999 634 Appendix Electrical and magnetic units Quantity S1 base units Mass Time Length Electric current Thermodynamic temperature Luminous intensity Amount of substance Derived units Potential difference, emf Resistance Electric charge Capacitance Conductance Magnetic flux Inductance Magnetic flux density Magnetic field strength Current density Electric field strength Permittivity Permeability Unit and symbol Derivation kilogram, kg second, s meter, m ampere, A kelvin, K candela, cd mole, mol vol, V ohm, ⍀ coulomb, C farad, F siemens, S weber, Wb henry, H tesla, T ampere per meter ampere per square meter volt per meter farad per meter henry per meter W и AϪ1 ϭ m2 и kg и sϪ3 и AϪ1 V и AϪ1 ϭ m2 и kg и sϪ3 и AϪ2 sиA C и VϪ1 ϭ mϪ2 и kgϪ1 и s4 и A2 A и VϪ1 ϭ mϪ2 и kgϪ1 и s3 и A2 V и s ϭ m2 и kg и sϪ2 и AϪ1 Wb и AϪ1 ϭ m2 и kg и sϪ2 и AϪ2 Wb и mϪ2 ϭ kg и sϪ2 и AϪ1 mϪ1 и A mϪ2 и A V и mϪ1 ϭ m и kg и sϪ3 и AϪ1 F и mϪ1 ϭ mϪ3 и kgϪ1 и s4 и A2 H и mϪ1 ϭ m и kg и sϪ2 и AϪ2 Dimensional formulas of common quantities Quantity Definition Mass Length Time Velocity Acceleration Force Momentum Energy Angle Angular velocity Angular acceleration Torque Angular momentum Moment of inertia Area Volume Density Pressure Action Viscosity Fundamental Fundamental Fundamental Distance/time Velocity/time Mass ϫ acceleration Mass ϫ velocity Force ϫ distance Arc/radius Angle/time Angular velocity/time Force ϫ lever arm Momentum ϫ lever arm Mass ϫ radius squared Length squared Length cubed Mass/volume Force/area Energy ϫ time Force per unit area per unit velocity gradient 635 Dimensional formula M L T LT Ϫ1 LT Ϫ2 MLT Ϫ2 MLT Ϫ1 ML2T Ϫ2 T Ϫ1 T Ϫ2 ML2T Ϫ2 ML2T Ϫ1 ML2 L2 L3 MLϪ3 MLϪ1T Ϫ2 ML2T Ϫ1 MLϪ1T Ϫ1 Appendix Internal energy and generalized work Type of energy Mechanical Expansion Stretching Extension Thermal Chemical Electrical Gravitational Polarization Electrostatic Magnetic Intensive factor Extensive factor Element of work Pressure (P) Surface tension (␥ ) Tensile stretch (F) Temperature (T) Chemical potential (gm) Electric potential (E) Gravitational field strength (mg) Volume (V) Area (A) Length (l) Entropy (S) Moles (n) Charge (Q) Height (h) ϪPdV ␥dA Fdl TdS dn EdQ mgdh Electric field strength (E) Total electric polarization (P) Total magetic polarization (M) EdP Magnetic field strength (H) 636 HdM Ύ f (u) du F(u) 637 n du ϭ u du ϭ Ύe du ϭ e u ϭ ln ȊuȊ u ln a du ϭ Ύ cos u du ϭ sin u Ύ sin u du ϭ Ϫcos u Ύa Ύe u du n Ϫ1 [for n ϭ Ϫ1] e u ln a au , ϭ ln a ln a a Ͼ 0, a if u Ͼ or ln(Ϫu) if u Ͻ unϩ1 , nϩ1 Ύ u ϭ ln u Ύu dx where u ϭ f (x) [integration by parts] Ύ F{ f (x)} dx ϭ Ύ F(u) du du ϭ Ύ f Ј(x) du Ύ f (ax) dx ϭ a Ύ u dv ϭ uv Ϫ Ύ v du Ύ (u Ϯ v Ϯ w Ϯ иии) dx ϭ Ύ u dx Ϯ Ύ v dx Ϯ Ύ w dx Ϯ иии Ύ af (x) dx ϭ a Ύ f (x) dx Ύ a dx ϭ ax General rules of integration* 2 2 2 Ύ tan u du ϭ ln sec u ϭ Ϫln cos u Ύ cot u du ϭ ln sin u u Ύ sec u du ϭ ln (sec u ϩ tan u) ϭ ln tan 2 ϩ u Ύ csc u du ϭ ln (csc u Ϫ cot u) ϭ ln tan Ύ sec u du ϭ tan u Ύ csc u du ϭ Ϫcot u Ύ tan u du ϭ tan u Ϫ u Ύ cot u du ϭ Ϫcot u Ϫ u u sin 2u Ύ sin u du ϭ Ϫ ϭ (u Ϫ sin u cos u) u sin 2u Ύ cos u du ϭ ϩ ϭ (u ϩ sin u cos u) Ύ sec u tan u du ϭ sec u Ύ csc u cot u du ϭ Ϫcsc u Ύ sinh u du ϭ cosh u Ύ cosh u du ϭ sinh u Ύ u du ϭ ln cosh u Ύ coth u du ϭ ln sinh u Appendix 638 Ϫ1 u Ϫ1 Ϫ1 u 2 2 2 2 2 (nϪ1) Ϫ1 (nϪ2) Ϫ1 2 uϪa 2 2 Ϫ1 2 Ϫ1 (nϪ3) u Ϫ1 2 This is called generalized integration by parts 2 du (n) 2 n 2 Ύ u Ϫ a ϭ 2a ln u ϩ a ϭ Ϫa coth a u Ͼ a 1 du aϩu u Ύ a Ϫ u ϭ 2a ln a Ϫ u ϭ a a u Ͻ a du u Ύ Ίa Ϫ u ϭ sin a du u Ύ Ίu ϩ a ϭ ln(u ϩ Ίu ϩ a ) or sinh a du Ύ Ίu Ϫ a ϭ ln(u ϩ Ίu Ϫ a ) du u Ύ uΊu Ϫ a ϭ a sec ͿaͿ du a ϩ Ίu ϩ a Ύ uΊu ϩ a ϭ Ϫa ln u du a ϩ Ίa Ϫ u Ύ uΊa Ϫ u ϭ Ϫa ln u Ύ f g dx ϭ f g Ϫ f gЈ ϩ f gЉ Ϫ иии (Ϫ1) Ύ fg (n) dx *Here, u, v, w are functions of x; a, b, p, q, n any constants, restricted if indicated; e ϭ 2.71828 is the natural base of logarithms; ln u denotes the natural logarithm of u where it is assumed that u Ͼ [in general, to extend formulas to cases where u Ͻ as well, replace ln u by ln ȊuȊ]; all angles are in radians; all constants of integration are omitted but implied SOURCE: Murray R Spiegel and John Liu, Mathematical Handbook of Formulas and Tables, 2d ed., Schaum’s Outline Series, McGraw-Hill, 1999 2 2 2 Ϫ1 Ύ sech u du ϭ sin (tanh u) or tan e u Ύ csch u du ϭ ln or Ϫcoth e Ύ sech u du ϭ u Ύ csch u du ϭ Ϫcoth u Ύ u du ϭ u Ϫ u Ύ coth u du ϭ u Ϫ coth u sinh 2u u Ύ sinh u du ϭ Ϫ ϭ 2(sinh u cosh u Ϫ u) sinh 2u u Ύ cosh u du ϭ ϩ ϭ 2(sinh u cosh u ϩ u) Ύ sech u u du ϭ Ϫsech u Ύ csch u coth u du ϭ Ϫcsch u du u Ύ u ϩ a ϭ a tan a General rules of integration* (cont.) Appendix Appendix Schematic electronic symblols* *From S Gibilisco, The Illustrated Dictionary of Electronics, 8th ed., McGraw-Hill, 2001 639 Appendix 640 Appendix 641 Appendix 642 Appendix 643 ... as listings of useful mathematical, engineering, and scientific data It is the editors’ hope that the Second Edition of the McGraw-Hill Dictionary of Engineering will serve the needs of scientists,... Terms of Use This page intentionally left blank Preface The McGraw-Hill Dictionary of Engineering provides a compendium of more than 18,000 terms that are central to the various branches of engineering. .. gear, from the coincidence of the pitch points of a pair of teeth until the last point of contact of the teeth { aŋиgəl əv re ses } ¯ angle of repose [ENG] See angle of rest [MECH] The angle between