## The Very Basics of TensorsTensor calculus is a generalization of vector calculus, and comes near of being a universal language in physics. Physical laws must be independent of any particular coordinate system used in describing them. This requirement leads to tensor calculus. The only prerequisites for reading this book are a familiarity with calculus (including vector calculus) and linear algebra, and some knowledge of differential equations. |

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### Contents

Spaces points and coordinates | 1 |

Distance between adjacent points in a Euclidean 2space for example an ordinary plane a Euclidean 3space for example ordinary space and a Euclidea... | 2 |

Flat and curved spaces | 3 |

Transformation of coordinates in Euclidean 3space Curvilinear coordinates | 4 |

Orthogonal curvilinear coordinates | 5 |

Vectors in Euclidean 3space with general curvilinear coordinates | 6 |

Contravariant and covariant components of a vector in Euclidean 3space | 10 |

The relation between the contravariant components of a vector in two general curvilinear coordinate systems in Euclidean 3space | 11 |

How to perform calculations | 46 |

The line element and the metric tensor Riemannian space | 49 |

The Jacobian of coordinate transformations in Euclidean 3space | 53 |

Area and volume in Euclidean 3space | 55 |

The Kronecker delta the Kronecker tensor | 61 |

The conjugate metric tensor the reciprocal metric tensor | 63 |

Associated tensors Raising and lowering indices | 66 |

The physical components of a vector Ap or A | 70 |

The relation between the covariant components of a vector in two general curvilinear coordinate systems in Euclidean 3space | 14 |

Proof of the result in section 8 Generalization to TVspace | 16 |

Proof of the result in section 9 | 19 |

General differentiable TVspaces and general coordinates Curves and surfaces | 23 |

Contravariant tensors of rank one | 25 |

Covariant tensors of rank one | 28 |

Mixed tensors of the second rank | 30 |

Contravariant and covariant tensors of rank two | 31 |

Tensors of rank greater than two | 33 |

Invariants or scalars tensors of rank zero | 34 |

Tensor equations | 35 |

Tensor fields | 36 |

Addition of tensors | 37 |

Subtraction of tensors | 38 |

Outer Multiplication of tensors | 39 |

Division of tensors | 40 |

Contraction of tensors | 41 |

Inner multiplication of tensors | 42 |

Quotient law of tensors | 43 |

Symmetry | 44 |

Skewsymmetry or antisymmetry | 45 |

The physical components of a tensor Apq or A | 72 |

Distance in a Riemannian space | 74 |

Angle between two curves in a Riemannian space | 76 |

Length of a vector in a Riemannian space | 78 |

Angle between two vectors in a Riemannian space | 79 |

Notes on spaces coordinates tensors and physics | 80 |

Partial derivative ordinary derivative of a tensor in general Aspace | 82 |

The Lie derivative of a general tensor field | 83 |

The affine connection and covariant differentiation | 89 |

The intrinsic absolute derivative and affine geodesies | 99 |

The RiemannChristoffel tensor | 102 |

Affine flatness | 105 |

Christoffels symbols | 107 |

Metric geodesies | 112 |

The metric connection the covariant derivative and the absolute derivativ | 121 |

Metric flatness | 123 |

A few easy examples from physics in a general Riemannian Aspace with general coordinates | 124 |

A few easy exercises | 129 |

131 | |

136 | |

### Common terms and phrases

called Cartesian coordinate system Christoffel symbol contravariant tensor coordinates in Euclidean coordinates xr covariant components covariant derivative covariant tensor curvilinear coordinate system d2xr define differentiable TV-space dr dr ds ds du2 du2 du2 du2 du3 du3 _ du3 dx duc duc duq dup duq dx dx dx dy dz dxj dxk dxrK dy dy dz du2 Euclidean 3-space example functions inner product invariant Lie derivative literal suffixes metric connection metric form metric geodesic metric tensor mixed tensor Note ordinary orthogonal curvilinear coordinates outer multiplication outer product parameter Proof quantities rank and type rectangular Cartesian coordinates Riemann tensor Riemann tensor vanishes Riemannian space Riemannian TV-space scalar summation tangent vector Tensor calculus tensor field tensor of rank theorem transformation equations transformation law transformation of coordinates vector field