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Contents

• V Full version
• 0 Introduction
• 1 Derivatives and coordinates
• 1.1 Derivative of functions
• 1.2 Inverse functions
• 1.3 Coordinate systems
• 2 Curves and Line
• 2.1 Parametrised curves, lengths and arc length
• 2.2 Line integrals of vector fields
• 2.3 Gradients and Differentials
• 2.4 Work and potential energy
• 3 Integration in ℝ² and ℝ²
• 3.1 Integrals over subsets of ℝ²
• 3.2 Change of variables for an integral in ℝ²
• 3.3 Generalization to ℝ³
• 3.4 Further generalizations
• 4 Surfaces and surface integrals
• 4.1 Surfaces and Normal
• 4.2 Parametrized surfaces and area
• 4.3 Surface integral of vector fields
• 4.4 Change of variables in ℝ² and ℝ² revisited
• 5 Geometry of curves and surfaces
• 6 Div, Grad, Curl and ∇
• 6.1 Div, Grad, Curl and ∇
• 6.2 Second-order derivatives
• 7 Integral theorems
• 7.1 Statement and examples
• 7.2 Relating and proving integral theorems
• 8 Some applications of integral theorems
• 8.1 Integral expressions for div and curl
• 8.2 Conservative fields and scalar products
• 8.3 Conservation laws
• 9 Orthogonal curvilinear coordinates
• 9.1 Line, area and volume elements
• 9.2 Grad, Div and Curl
• 10 Gauss' Law and Poisson's equation
• 10.1 Laws of gravitation
• 10.2 Laws of electrostatics
• 10.3 Poisson's Equation and Laplace's equation
• 11 Laplace's and Poisson's equations
• 11.1 Uniqueness theorems
• 11.2 Laplace's equation and harmonic functions
• 11.3 Integral solutions of Poisson's equations
• 12 Maxwell's equations
• 12.1 Laws of electromagnetism
• 12.2 Static charges and steady currents
• 12.3 Electromagnetic waves
• 13 Tensors and tensor fields
• 13.1 Definition
• 13.2 Tensor algebra
• 13.3 Symmetric and antisymmetric tensors
• 13.4 Tensors, multi-linear maps and the quotient rule
• 13.5 Tensor calculus
• 14 Tensors of rank 2
• 14.1 Decomposition of a second-rank tensor
• 14.2 The inertia tensor
• 14.3 Diagonalization of a symmetric second rank tensor
• 15 Invariant and isotropic tensors
• 15.1 Definitions and classification results
• 15.2 Application to invariant integrals