MTH622 Complete Handouts Modules 1-186

i

About the Handouts

The following books have been mainly followed to prepare the slides and handouts: 1. Spiegel, M.R., Theory and Problems of Vector Analysis: And an Introduction to Tensor Analysis. 1959: McGraw-Hill. 2. Spiegel, M.S., Theory and problems of theoretical mechanics. 1967: Schaum. 3. Taylor, J.R., Classical Mechanics. 2005: University Science Books. 4. DiBenedetto, E., Classical Mechanics: Theory and Mathematical Modeling. 2010: Birkhäuser Boston. 5. Fowles, G.R. and G.L. Cassiday, Analytical Mechanics. 2005: Thomson Brooks/Cole. The first two books were considered as main text books. Therefore the students are advised to read the first two books in addition to these handouts. In addition to the above mentioned books, some other reference book and material was used to get these handouts prepared. ii

Contents

Module No. 1 .......................................................................................................................................... 1

Module No. 2 .......................................................................................................................................... 2

Module No. 3 .......................................................................................................................................... 4 The Operator Del and Gradient of Function .................................................................................... 4 Module No. 4 .......................................................................................................................................... 6 Properties of the Gradient ................................................................................................................. 6 Module No. 5 .......................................................................................................................................... 8 Directional Derivative........................................................................................................................ 8 Module No. 6 ........................................................................................................................................ 10 Theorem related Directional Derivative ......................................................................................... 10 Module No. 7 ........................................................................................................................................ 11 Example of the Directional Derivative ............................................................................................ 11 Module No. 8 ........................................................................................................................................ 12 Related Problem 1 of the Directional Derivative ............................................................................ 12 Module No. 9 ........................................................................................................................................ 13 Related problem 2 of Directional Derivative .................................................................................. 13 Module No. 10 ...................................................................................................................................... 14 Related Problem 3 of the Directional Derivative ............................................................................ 14 Module No. 11 ...................................................................................................................................... 16 Geometrical Interpretation of Gradient ......................................................................................... 16 Module No. 12 ...................................................................................................................................... 17 Theorem Related to Gradient.......................................................................................................... 17 Module No. 13 ...................................................................................................................................... 19 Related Problem 1; Gradient .......................................................................................................... 19 Module No. 14 ...................................................................................................................................... 20 Related Problem 2; Gradient .......................................................................................................... 20 Module No. 15 ...................................................................................................................................... 21 Related Problem 3; Gradient .......................................................................................................... 21 Module No. 16 ...................................................................................................................................... 23 Divergence of a Vector Point Function ........................................................................................... 23 iii

Module No. 17 ...................................................................................................................................... 25 Properties of the Divergence ........................................................................................................... 25 Module No. 18 ...................................................................................................................................... 27 Laplacian ......................................................................................................................................... 27 Module No. 19 ...................................................................................................................................... 29 Example of Divergence .................................................................................................................... 29 Module No. 20 ...................................................................................................................................... 30 Related Problem 1: Divergence ....................................................................................................... 30 Module No. 21 ...................................................................................................................................... 33 Related Problem 2: Divergence ....................................................................................................... 33 Module No. 22 ...................................................................................................................................... 35 Related Problem 3: Laplacian ......................................................................................................... 35 Module No. 23 ...................................................................................................................................... 37 Curl of a vector Point Function ....................................................................................................... 37 Module No. 24 ...................................................................................................................................... 39 Properties of the Curl ...................................................................................................................... 39 Module No. 25 ...................................................................................................................................... 42 Example of Curl............................................................................................................................... 42 Module No. 26 ...................................................................................................................................... 43 Related Problem 1: Curl.................................................................................................................. 43 Module No. 27 ...................................................................................................................................... 45 Related Problem 2: Curl.................................................................................................................. 45 Module No. 28 ...................................................................................................................................... 48 Related Problem 3: Curl.................................................................................................................. 48 Module No. 29 ...................................................................................................................................... 50 Vector Identities............................................................................................................................... 50 Module No. 30 ...................................................................................................................................... 54 Line Integral .................................................................................................................................... 54 Module No. 31 ...................................................................................................................................... 56 Other forms and General Properties of line Integrals .................................................................... 56 Module No. 32 ...................................................................................................................................... 58

Module No. 33 ...................................................................................................................................... 60