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CORNELL UNIVERSITY Department of Theoretical * Applied Mechanics The College of Engineering- Thuraton Kail CORNELL UNIVERSITY Ithaca, N. Y. 14850 ENGINEERING MATHEMATICS Volume 1 Ithaca, New York H' u- Bl0Ck E. T. Cranch P. J. Hilton R. J. Walker E N G I N E E R I N G M A T H E M A T I C S Volume 1. by H. D. Block Professor of Mechanics E. T. Cranch Professor of Mechanics P. J. Hilton Professor of Mathematics R. J. Walker Professor of Mathematics CORNELL UNIVERSITY Fifth Printing The preparation of this material has been partially supported by- Grant NSF-GE2564 made to Cornell Uni versity by the National Science Foun dation. Permission will be granted by the copyright owner to all persons to make any use of this work after August 31? 196?; provided (l) that notice be given to the copyright owner before such use is made, (2) that publications incorpo rating materials covered by this copyright contain an acknowledgement of this copyright and a statement that the publication is no,t endorsed by the copyright holder or the original authors and (3) that whenever materials covered by this copyright are represented to be the work of the original authors, they shall be republished without charge. Copyright 196k by Cornell University Ithaca, Tlew York Preface These two volumes are the basis of the sophomore mathematics program for the students in the College of Engineering at Cornell University. It is not expected that all this material will be covered by all the students in the seven semester hours allotted to this portion of the curriculum* There is room for some selection of topics, and in particular the more difficult topics and problems are intended only for the best students- The freshman year contains eight semester hours of mathematics - essentially the first fifteen chapters of Calculus and Analytic Geometry, by C. B. Thomas, Jr., (Addison-Wesley Publishing Co., Reading, Mass., Third Edition, i960). In additon to this mathematical background the fresh man student learns how to program in the Cornell Computing Language (CORC) and how to have his problems run in the Computing Center. Com pletion of most of the classical topics of differential and integral calculus in the freshman year permits a new approach to the mathematics of the sophomore year. The sophomore program contains features that we believe will develop into significant advances in mathematical education. The arrangement of courses, syllabi, and the interdisciplinary nature of the effort are in accord with contemporary and future developments in the undergraduate mathematics education of engineers and scientists.* The dominant features include: * See "Recommendations on the Undergraduate Mathematics Program for Engineers and Physicists” Report Number 5> January 1962, Committee on the Undergradu ate Program in Mathematics, Mathematical Association of America. 1 . The opportunity to interweave physical interpretation and. engi neering applications, hence stirring the students' imagination and motivation and reducing the separation between theory and practice. 2. A substantial period of time devoted to linear transformations and matrices, linear equations, quadratic forms and elementary eigenvalue theory. 3. The introduction of vector field theory in the sophomore year so that students will be prepared for the field theory approach in junior year applied science courses in fluid and solid me chanics, transport phenomena, and electromagnetism. b. The inclusion of numerical and machine techniques in such mathe matical topics as differential equations, infinite series, and linear algebra. 5. An interdisciplinary effort at both the planning and working levels whereby professional mathematicians and engineers create a course having balance between mathematical rigor and sophis tication and physical insight and motivation. 6. Coordination of mathematics with the sophomore mechanics and electrical science courses. 7. A mathematics program upon which the student will "be encouraged to build in subsequent years. If a sophomore course constructed along these lines proves success ful it is hoped to extend the program at a later date to include a rewor king of the freshman material, so as eventually to provide a coordinated program for the first two years of engineering mathematics. Of the many people who helped us we wish to thank particularly our colleagues Professor G. R. Livesay and Professor N. DeClaris, who were heavily involved in the initial planning; Mrs, Sandria Kerr, Mr. Neal Plotkin, Mr. Terry Gardner and Mr. Frank Moon, graduate students who served as critics and editors; Miss. Virginia Cranch, who did most of the final typing; and Mr. Harry Gleason, our most obliging printer. Our greatest thanks are due to the National Science Foundation for its generous support. H. D. Block E. T. Cranch P. J. Hilton R. J. Walker Ithaca, New York September 196^+ Differential Equations Table of Contents lo Introduction. . . . ............................. 1.1 2o The Fundamental Theorem of the First Order Equation - Euler’s Method................. . ... 1.10 3. Systems of First Order Simultaneous Equations.. .... 1.24 ho Higher Order Differential Equations.................1*37 5o Special Integrable Forms...........- ............. 1.46 6. The First Order Linear Equation.................... 1.64 7- Applications. ......... ....................... . l„8o Line.ar Algebra and. Matrix Theory Table of Contents 1. Solving a System of Linear Equations with Numeri cal Coefficients by the Method of Elimination.. . ... .2.1 VECTOR SPACES 2. Physical Vectors................................. 2.25 3. Vector Spaces........................... . . . . 2.29 4. Further Properties of Physical Vectors. .......... . 2.40 5. Vector Subspaces. Linear Combination...............2.43 6. Linear Dependence................................ 2.5.1 7. Bases. Dimension of a Vector Space................ 2.59 7A. The Steinitz Replacement Theorem....................2.65 8. Isomorphism. Change of Basis......................2.69 LINEAR TRANSFORMATIONS AND MATRICES 9. Examples of Linear Transformations. . . ............. 2.76 10. Multiplication of Matr ice s................ .... 2.85 11. Solution of Linear Equations.......... ........... 2.95 12. Matrix Equations. Inverse Matrices................ 2.1.12 13. Determinants. . . . . . . . . . ....................2.124 14. Cofactors. Cramer's Rule......................... 2.139 15. Linear Transformations................. ......... 2.154 16. Miscellaneous Problems and Applications..............2.176 Infinite Series Table of Contents 1. Cutting Up a Cheesecake.......................... 3-1 2. Convergence of a Series of Constants................3*8 3o The Generous Donor................................3*22 ko The Geometric Series............................. 3*25 TAYLOR SERIES 5o Power Series.................................... 3*38 6. Representation of Functions-by Power Series.......... 3*48 7o Taylor’s Formula for the Remainder........... . 3-53 8. Taylor Series.. ............................ . 3*64 9» Taylor Series Solutions of Differential Equations.. . . 3*68 10. Indeterminate Forms............... .............. 3*7^ CONVERGENCE OF SERIES 11c Basic Facts about Series and Sequences.............. 3-80 12 o Absolute and Conditional Convergence................3*82 13• Comparison Test.. . . . . . . . . . ................ 3-83 l4„ Ratio Test. . . . . . . ......... ................ 3*88 15. Integral Test.. ................ ................ 3-95 16. Alternating Series with Decreasing Terms.......... . 3*98 17o Uniform Convergence.. . . . . . . . . . ......... . . 3*102