# Part IA - Vector Calculus

## Lectured by B. Allanach, Lent 2015

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

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# Contents

- V Full version
- 0 Introduction
- 1 Derivatives and coordinates
- 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 ℝ
^{2}and ℝ^{2} - 3.1 Integrals over subsets of ℝ
^{2} - 3.2 Change of variables for an integral in ℝ
^{2} - 3.3 Generalization to ℝ
^{3} - 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 ℝ
^{2}and ℝ^{2}revisited - 5 Geometry of curves and surfaces
- 6 Div, Grad, Curl and ∇
- 7 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
- 10 Gauss' Law and Poisson'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
- 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