Assume a Cartesian coordinate system in which , and are unit vectors in the three coordinate directions. is defined such that:

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1.1.4.1.1. Gradient Operator

For a general scalar function , the gradient of is defined by:

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1.1.4.1.2. Divergence Operator

For a vector function where:

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the divergence of is defined by:

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1.1.4.1.3. Dyadic Operator

The dyadic operator (or tensor product) of two vectors, and , is defined as:

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By using specific tensor notation, the equations relating to each dimension can be combined into a single equation. Thus, in the specific tensor notation:

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The transpose of a matrix is defined by the operator . For example, if the matrix is defined by:

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then:

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The Identity matrix is defined by:

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Although index notation is not generally used in this documentation, the following may help you if you are used to index notation.

In index notation, the divergence operator can be written:

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where the summation convention is followed; that is, the index is summed over the three components.

The quantity can be represented by (when and are vectors), or by (when is a vector and is a matrix), and so on.

Hence, the quantity can be represented by:

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Note the convention that the derivatives arising from the divergence operator are derivatives with respect to the same coordinate as the first listed vector. That is, the quantity is represented by:

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and not:

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The quantity (when and are matrices) can be written by .