You have some good examples, mixed with customary oddities like (see Tangent
Vector reference)
T(t) -: r' / |r'| Formula (1), explicit on left, tacit on right
T(t) = dr / ds Formula (3), left "function of t", right "function of s"
Your point is valid, tacit usage does occur in advanced mathematics.
"Consistently used part of the time" captures the flavor!
Oleg Kobchenko wrote:
From: Kip Murray <[email protected]>
TACIT DEFINITION OF VERBS DEFINES VERBS IN TERMS OF OTHER VERBS. This occurs in
mathematics where you see h = f + g , h = f g, h = f o g, etc., but tacit
definition is not much used in math beyond differentiation formulas like (f +
g)' = f' + g', (f g)' = f' g + f g', (f o g)' = (f' o g) g' ; and you are more
than likely to see the last expressed as f(g(x))' = f'(g(x)) g'(x) . Likewise,
you are more likely to see (x^n)' = n x^(n-1) than (id^n)' = n id^(n-1) . That
is, math is more likely to use informal definitions with x's than tacit
definitions without x's.
Tacit is not much used beyond differentiation formulas?
How about Functional Analysis:
http://mathworld.wolfram.com/L2-Space.html
Convolution (half the formulas are tacit)
http://mathworld.wolfram.com/Convolution.html
Vector Algebra and Differential Geometry
http://mathworld.wolfram.com/RadiusVector.html
http://mathworld.wolfram.com/TangentVector.html
I'd say it is consistently used part of the time after the
nature of the arguments is presented and it facilitates
shorter and cleaner notation.
In general it is used whenever function itself is an object
of transformation or calculation, such as whenever operators
appear. Differentiation is just one of them.
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