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Vector Calculus

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Table of Contents

1. Vector Algebra.- 1.1 Vectors and scalars.- 1.1.1 Definition of a vector and a scalar.- 1.1.2 Addition of vectors.- 1.1.3 Components of a vector.- 1.2 Dot product.- 1.2.1 Applications of the dot product.- 1.3 Cross product.- 1.3.1 Applications of the cross product.- 1.4 Scalar triple product.- 1.5 Vector triple product.- 1.6 Scalar fields and vector fields.- 2. Line, Surface and Volume Integrals.- 2.1 Applications and methods of integration.- 2.1.1 Examples of the use of integration.- 2.1.2 Integration by substitution.- 2.1.3 Integration by parts.- 2.2 Line integrals.- 2.2.1 Introductory example: work done against a force.- 2.2.2 Evaluation of line integrals.- 2.2.3 Conservative vector fields.- 2.2.4 Other forms of line integrals.- 2.3 Surface integrals.- 2.3.1 Introductory example: flow through a pipe.- 2.3.2 Evaluation of surface integrals.- 2.3.3 Other forms of surface integrals.- 2.4 Volume integrals.- 2.4.1 Introductory example: mass of an object with variable density.- 2.4.2 Evaluation of volume integrals.- 3. Gradient, Divergence and Curl.- 3.1 Partial differentiation and Taylor series.- 3.1.1 Partial differentiation.- 3.1.2 Taylor series in more than one variable.- 3.2 Gradient of a scalar field.- 3.2.1 Gradients, conservative fields and potentials.- 3.2.2 Physical applications of the gradient.- 3.3 Divergence of a vector field.- 3.3.1 Physical interpretation of divergence.- 3.3.2 Laplacian of a scalar field.- 3.4 Curl of a vector field.- 3.4.1 Physical interpretation of curl.- 3.4.2 Relation between curl and rotation.- 3.4.3 Curl and conservative vector fields.- 4. Suffix Notation and its Applications.- 4.1 Introduction to suffix notation.- 4.2 The Kronecker delta ?ij.- 4.3 The alternating tensor ?ijk.- 4.4 Relation between ?ijk and ?ij.- 4.5 Grad, div and curl in suffix notation.- 4.6 Combinations of grad, div and curl.- 4.7 Grad, div and curl applied to products of functions.- 5. Integral Theorems.- 5.1 Divergence theorem.- 5.1.1 Conservation of mass for a fluid.- 5.1.2 Applications of the divergence theorem.- 5.1.3 Related theorems linking surface and volume integrals.- 5.2 Stokes's theorem.- 5.2.1 Applications of Stokes's theorem.- 5.2.2 Related theorems linking line and surface integrals.- 6. Curvilinear Coordinates.- 6.1 Orthogonal curvilinear coordinates.- 6.2 Grad, div and curl in orthogonal curvilinear coordinate systems.- 6.2.1 Gradient.- 6.2.2 Divergence.- 6.2.3 Curl.- 6.3 Cylindrical polar coordinates.- 6.4 Spherical polar coordinates.- 7. Cartesian Tensors.- 7.1 Coordinate transformations.- 7.2 Vectors and scalars.- 7.3 Tensors.- 7.3.1 The quotient rule.- 7.3.2 Symmetric and anti-symmetric tensors.- 7.3.3 Isotropic tensors.- 7.4 Physical examples of tensors.- 7.4.1 Ohm's law.- 7.4.2 The inertia tensor.- 8. Applications of Vector Calculus.- 8.1 Heat transfer.- 8.2 Electromagnetism.- 8.2.1 Electrostatics.- 8.2.2 Electromagnetic waves in a vacuum.- 8.3 Continuum mechanics and the stress tensor.- 8.4 Solid mechanics.- 8.5 Fluid mechanics.- 8.5.1 Equation of motion for a fluid.- 8.5.2 The vorticity equation.- 8.5.3 Bernoulli's equation.- Solutions.

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P.C. Matthews Vector Calculus "Written for undergraduate students in mathematics, the book covers the material in a comprehensive but concise manner, combining mathematical rigor with physical insight. There are many diagrams to illustrate the physical meaning of the mathematical concepts, which essential for a full understanding of the subject." - ZENTRALBLATT MATH

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