An Introduction to Vector Analysis

1 Definitions. Addition of Vectors.- 1. Scalar and Vector Quantities.- 2. Graphical Representation of Vectors.- 3. Addition and Subtraction of Vectors.- 4. Components of a Vector.- 5. Geometrical Applications.- 6. Scalar and Vector Fields.- Miscellaneous Exercises I.- 2 Products of Vectors.- 1. General.- 2. The Scalar Product.- 3. The Vector Product.- 4. Vector Area.- 5. Application to Vector Products.- 6. Products of Three Vectors.- 7. Line and Surface Integrals as Scalar Products.- Miscellaneous Exercises II.- 3 The Differentiation of Vectors.- 1. Scalar Differentiation.- 2. Differentiation of Sums and Products.- 3. Partial Differentiation.- Miscellaneous Exercises III.- 4 The Operator ? and Its Uses.- 1. The Operator ?.- 2. The Gradient of a Scalar Field.- 3. The Divergence of a Vector Field.- 4. The Operator div grad..- 5. The Operator ?2 with Vector Operand.- 6. The Curl of a Vector Field.- 7. Simple Examples of the Curl of a Vector Field.- 8. Divergence of a Vector Product.- 9. Divergence and Curl of SA.- 10. The Operator curl grad..- 11. The Operator grad div..- 12. The Operator div curl..- 13. The Operator curl curl..- 14. The Vector Field grad (k/r).- 15. Vector Operators in Terms of Polar Co-ordinates.- Miscellaneous Exercises IV.- 5 Integral Theorems.- 1. The Divergence Theorem of Gauss.- 2. Gauss's Theorem and the Inverse Square Law.- 3. Green's Theorem.- 4. Stokes's Theorem.- 5. Alternative Definitions of Divergence and Curl.- 6. Classification of Vector Fields.- Miscellaneous Exercises V.- 6 The Scalar Potential Field.- 1. General Properties.- 2. The Inverse Square Law. Point Sources.- 3. Volume Distributions.- 4. Multi-valued Potentials.- 7 The Vector Potential Field.- 1. The Magnetic Field of a Steady Current.- 2. The Vector Potential.- 3. Linear Currents.- 4. Simple Examples of Vector Potential.- 8 The Electromagnetic Field Equations of Maxwell.- 1. General.- 2. Maxwell's Equations.- 3. Energy Considerations.- Miscellaneous Exercises VIII.- Answers to Exercises.