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"Suppose $\phi$ is a function of the coordinates $x^k$, and let denote $\bar{\phi}$ the functional value under a transformation to a new set of coordinates $\bar{x}^k$. Then $\phi$ is called a scalar or invariant with respect to the coordinate transformation if $\phi=\bar{\phi}$. A scalar or invariant is also called a tensor of rank zero."

Maybe this is obvious but the term "functional value" has not been defined anywhere in the book. I don't know what it means and therefore don't understand the definition.

The definition is from "Schaum's Outline of Vector Analysis", chapter 8.

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1 Answer

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A functional is a function that takes a vector as input and gives a number as output. A functional value is the value of the functional when you plug in a vector. They are saying that if you get the same number in different coordinate systems, it is a "tensor of rank zero."

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