By James G. Simmonds
During this textual content which progressively develops the instruments for formulating and manipulating the sector equations of Continuum Mechanics, the maths of tensor research is brought in 4, well-separated phases, and the actual interpretation and alertness of vectors and tensors are under pressure all through. This re-creation includes extra workouts. additionally, the writer has appended a piece on Differential Geometry
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As shown, the uncertainties in position through observation and forecast overlap, a Venn diagram of sorts. It is these uncertainties, generally expressed as error variances, that determine the estimate. 5 Stochastic/Dynamic problem 35 We demonstrate with the following strategy. , X = w 1 X f + w2 z where the error variance of the forecast is σf2 and the error variance of the observation is σz2 . , w1 + w2 = 1, or restated in terms of weights W and (1 − W ), X = W X f + (1 − W )z. If X T is the true position at the next time of observation/forecast, we can rewrite the expression as X − X T = W (X f − X T ) + (1 − W )(z − X T ) or ε = W εf + (1 − W )εz where ε, εf , and εz are errors associated with the estimate, the forecast, and the observation respectively, where (ε)2 = σ 2 , (ε f )2 = σf2 , (εz )2 = σz2 the overbar indicating ensemble variance or variance over many trials.
In the case where the estimate is formed from the analysis at t = 0 and the forecasts from −T, −2T, . . 6) i=1 √ and the variance of the estimate would be ε 2 /N , or an rms error of ε/ N , the wellknown result from statistics; namely the error of the mean is reduced by a factor of square root of the number of members in the sample (“law of large numbers”). What has tacitly been assumed in this formulation is that the forecast error is the same no matter the duration of forecast. That is, the forecast error does not grow with time – the error at the initial time, an error related to the observational error, is propagated forward but does not grow.
A number of observations concerning these classiﬁcations are in order. (1) Discrete vs. , Newton’s laws of motion, laws of thermodynamics, laws of electromagnetics, and other generative and dissipative forces including absorption, emission, radiation, conduction, convection, evaporation, condensation, and turbulence. These equations by their very nature are continuous functions of time and are expressed as a system of ordinary or partial differential equations involving space and time variables.
A brief on tensor analysis by James G. Simmonds