Tensor Fields

Most tensor fields encountered in physics are smoothly varying and differentiable. Consider the first-order tensor field . The various partial derivatives of the components of this field with respect to the Cartesian coordinates are written

(B.50) |

Moreover, this set of derivatives transform as the components of a second-order tensor. In order to demonstrate this, we need the transformation rule for the , which is the same as that for a first-order tensor: that is,

(B.51) |

Thus,

(B.52) |

It is also easily shown that

Now,

(B.54) |

which is the correct transformation rule for a second-order tensor. Here, use has been made of the chain rule, as well as Equation (B.53). [Note, from Equation (B.26), that the are not functions of position.] It follows, from the previous argument, that differentiating a tensor field increases its order by one: for instance, is a third-order tensor. The only exception to this rule occurs when differentiation and contraction are combined. Thus, is a first-order tensor, because it only contains a single free index.

The gradient (see Section A.18) of a scalar field is an example of a first-order tensor field (i.e., a vector field):

(B.55) |

The divergence (see Section A.20) of a vector field is a contracted second-order tensor field that transforms as a scalar:

(B.56) |

Finally, the curl (see Section A.22) of a vector field is a contracted fifth-order tensor that transforms as a vector

(B.57) |

The previous definitions can be used to prove a number of useful results. For instance,

(B.58) |

which follows from symmetry because whereas . Likewise,

(B.59) |

which again follows from symmetry. As a final example,

(B.60) |

According to the divergence theorem (see Section A.20),

(B.61) |

where is a closed surface surrounding the volume . The previous theorem is easily generalized to give, for example,

(B.62) |

or

(B.63) |

or even

(B.64) |