Vector Analysis: A Physicist's Guide to the Mathematics of Fields in Three Dimensions
Vector analysis provides the language that is needed for a precise quantitative statement of the general laws and relationships governing such branches of physics as electromagnetism and fluid dynamics. The account of the subject is aimed principally at physicists but the presentation is equally appropriate for engineers. The justification for adding to the available textbooks on vector analysis stems from Professor Kemmer's novel presentation of the subject developed through many years of teaching, and in relating the mathematics to physical models. While maintaining mathematical precision, the methodology of presentation relies greatly on the visual, geometric aspects of the subject and is supported throughout the text by many beautiful illustrations that are more than just schematic. A unification of the whole body of results developed in the book - from the simple ideas of differentiation and integration of vector fields to the theory of orthogonal curvilinear coordinates and to the treatment of time-dependent integrals over fields - is achieved by the introduction from the outset of a method of general parametrisation of curves and surfaces.
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Summary of vector algebra
The geometrical background to vector analysis
Metric properties of Euclidean space
Scalar and vector fields
Further spatial integrals
Second derivatives of vector fields elements
Boundary behaviour of fields
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applied boundary curve boundary surface calculation cartesian coordinates chapter clearly closed curve closed surface const constant vector curl curl curlg curvilinear coordinates definition derivatives described differential dipole direction discontinuity discussion divergence theorem double layer equation Evaluate expression field f field line picture fluid dynamics flux integral generalisation given grad gradient field Hence identity integrand involving irrotational field level surface limit mathematical normal component notation Note origin orthogonal parameters parametrisation particular perpendicular physical plane Poisson's equation polar coordinates potential theory prove quantity quasi-cube quasi-square region of space replaced result scalar field scalar potential Show simple sin2 singularities solenoidal field solution source density sphere spherical polars Stokes stream function surface integral surface sources symmetry T-field three vectors unit vector values vanish vector analysis vector area vector field vector potential Verify visualise volume vorticity z-axis zero