Application of The Total Differential To Error Estimation
In measurement, the total differential is used in estimating the error Δf of a function f based on the errors Δx, Δy, ... of the parameters x, y, .... Assuming that the interval is short enough for the change to be approximately linear:
- Δf(x) = f'(x) × Δx
and that all variables are independent, then for all variables,
This is because the derivative fx with respect to the particular parameter x gives the sensitivity of the function f to a change in x, in particular the error Δx. As they are assumed to be independent, the analysis describes the worst-case scenario. The absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign. From this principle the error rules of summation, multiplication etc. are derived, e.g.:
- Let f(a, b) = a × b;
- Δf = faΔa + fbΔb; evaluating the derivatives
- Δf = bΔa + aΔb; dividing by f, which is a × b
- Δf/f = Δa/a + Δb/b
That is to say, in multiplication, the total relative error is the sum of the relative errors of the parameters.
Read more about this topic: Total Derivative
Famous quotes containing the words application of the, application of, application, total, differential, error and/or estimation:
“Most people, no doubt, when they espouse human rights, make their own mental reservations about the proper application of the word “human.””
—Suzanne Lafollette (1893–1983)
“It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct application of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”
—Jorge Luis Borges (1899–1986)
“I think that a young state, like a young virgin, should modestly stay at home, and wait the application of suitors for an alliance with her; and not run about offering her amity to all the world; and hazarding their refusal.... Our virgin is a jolly one; and tho at present not very rich, will in time be a great fortune, and where she has a favorable predisposition, it seems to me well worth cultivating.”
—Benjamin Franklin (1706–1790)
“I find myself ... hoping a total end of all the unhappy divisions of mankind by party-spirit, which at best is but the madness of many for the gain of a few.”
—Alexander Pope (1688–1744)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“Truth on our level is a different thing from truth for the jellyfish, and there must certainly be analogies for truth and error in jellyfish life.”
—T.S. (Thomas Stearns)
“No man ever stood lower in my estimation for having a patch in his clothes; yet I am sure that there is greater anxiety, commonly, to have fashionable, or at least clean and unpatched clothes, than to have a sound conscience.”
—Henry David Thoreau (1817–1862)