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2012

Differential Calculus, Tensor Products and the Importance of Notation

Manton, Jonathan H.

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An efficient coordinate-free notation is elucidated for differentiating matrix expressions and other functions between higher-dimensional vector spaces.

This method of differentiation is known, but not explained well, in the literature.
Teaching it early in the curriculum would avoid the tedium of element-wise differentiation and provide a better footing for understanding more advanced applications of calculus.
Additionally, it is shown to lead naturally to tensor products, a topic previously considered too difficult to motivate quickly in elementary ways.

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