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:
“By an application of the theory of relativity to the taste of readers, to-day in Germany I am called a German man of science, and in England I am represented as a Swiss Jew. If I come to be regarded as a bête noire the descriptions will be reversed, and I shall become a Swiss Jew for the Germans and a German man of science for the English!”
—Albert Einstein (1879–1955)
“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)
“My business is stanching blood and feeding fainting men; my post the open field between the bullet and the hospital. I sometimes discuss the application of a compress or a wisp of hay under a broken limb, but not the bearing and merits of a political movement. I make gruel—not speeches; I write letters home for wounded soldiers, not political addresses.”
—Clara Barton (1821–1912)
“The human spirit is itself the most wonderful fairy tale that can possibly be. What a magnificent world lies enclosed within our bosoms! No solar orbit hems it in, the inexhaustible wealth of the total visible creation is outweighed by its riches!”
—E.T.A.W. (Ernst Theodor Amadeus Wilhelm)
“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)
“Knowledge is ancient error reflecting on its youth.”
—Francis Picabia (1878–1953)
“... it would be impossible for women to stand in higher estimation than they do here. The deference that is paid to them at all times and in all places has often occasioned me as much surprise as pleasure.”
—Frances Wright (1795–1852)