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Engineering LibreTexts

4: Tensor Calculus

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    Figure \(\PageIndex{1}\): Vector field representation of the wind over the northwest Pacific ocean. The curl of this vector field is a dominant influence on ocean currents. Its divergence can tell us about vertical motion and precipitation (Ansley Manke, NOAA).

    A field is (for our purposes) a quantity that varies in space and can therefore be differentiated with respect to position. Scalars, vectors and tensors can all be fields (e.g., figure 4.1). The various derivative operations that can be applied to fields are fundamental tools in the study of flow. I assume that the reader is comfortable with the calculus as applied to functions of a single variable and has some familiarity with partial derivatives. With this knowledge, it is straightforward to apply the calculus to scalar, vector and tensor fields

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