A covariant tensor is a Tensor having specific transformation properties (c.f., a Contravariant Tensor). To
examine the transformation properties of a covariant tensor, first consider the Gradient

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

Contravariant Tensors are a type of Tensor with differing transformation properties,
denoted . However, in 3-D Cartesian Coordinates,

(7) |

(8) |

To turn a Contravariant Tensor into a covariant tensor, use the Metric Tensor to write

(9) |

**References**

Arfken, G. ``Noncartesian Tensors, Covariant Differentiation.'' §3.8 in
*Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 158-164,
1985.

Morse, P. M. and Feshbach, H. *Methods of Theoretical Physics, Part I.*
New York: McGraw-Hill, pp. 44-46, 1953.

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1999-05-25