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Introduction to Environmental Engineering Calculations
Published in Danny D. Reible, Fundamentals of Environmental Engineering, 2017
Differentiating this equation with respect to the best estimate μest and setting the result equal to zero gives the minimum value of the error variance, or the best least square estimate. This recognizes the rule of calculus that the zeros of the differential of a function determine its minima, maxima, and inflection points. The sign of the second derivative indicates which, in this case, a minima. Differentiating, recognizing that the sums represented by the first two terms are simply constants, () 0=−2∑xι+2NX¯est
Mathematical Preliminaries
Published in R. Ravi, Chemical Engineering Thermodynamics, 2020
Two questions bring out the confusion about differentials in a stark manner: Is the differential of a function of a single variable also a function of one variable alone?Does the differential of the function y = f(x) equal a small change in the variable y to a small change in the variable x?
Constitutive models
Published in Paulo B. Lourenço, Angelo Gaetani, Finite Element Analysis for Building Assessment, 2022
Paulo B. Lourenço, Angelo Gaetani
The yield surfaces are defined such as f(σ) = 0 and Eq. (3.2) provides more insight. According to differential calculus, this function can be increasing, decreasing or stationary based on the sign of the yield function rate f˙(σ), derived in Eq. (3.3). In the present section, rate measures the variability of a certain function with respect to time. In this case, df=f˙(σ)dt, where dt is the differential of time. The reader should not be misled because time is here introduced only with the purpose of ordering the events and no dynamic effects are considered. From the computational point of view, the differential quantities indicate, in reality, small increments associated with the load step of an increment loading process, as discussed in Chapter 2. Looking at Eq. (3.4), from the mechanical point of view, f˙(σ)>0 means that the value of f(σ) is increasing and approaching the yielding surface f(σ) = 0, i.e. the material is undergoing a loading process. Analogously, f˙(σ)<0 is indicative of unloading, and a null value represents the permanency of the elastic or plastic state.
Tracking control of soft dielectric elastomer actuator based on nonlinear PID controller
Published in International Journal of Control, 2022
Peng Huang, Jundong Wu, Chun-Yi Su, Yawu Wang
The reasons for the above settings are as follows. For the proportional control, the function produces a high output when the error is small, which is beneficial to improve the rapidity of the closed-loop control system and facilitates to mitigate the phase-delay caused by the hysteresis nonlinearity of the SDEA. On the other hand, the function produces a low output when the error is large, which is conductive to prevent the high frequency chattering of the SDEA caused by the excessive output. For the integral control and differential control, the function has similar effects. Besides, for the integral control, the function is beneficial to handle the integral windup problem encountered in practical experiments (Gao et al., 2001). More importantly, for the differential control, the function facilitates to produce the favourable differential action in practical experiments of the SDEA. Under the regulation of the differential control, the NEFC can predict the trend of the error and provide the phase-lead compensation to reduce the phase-delay of the SDEA.
A pair of linear canonical Hankel transformations and associated pseudo-differential operators
Published in Applicable Analysis, 2018
Further,The function is a solution for the differential equation The function is a solution for the differential equation
Intra-mathematical connections made by high school students in performing Calculus tasks
Published in International Journal of Mathematical Education in Science and Technology, 2018
Javier García-García, Crisólogo Dolores-Flores
It is important to know the differential of a function before integrating because it will allow us to decide with respect to which variable we must integrate if there is more than one variable in the function. Students established one connection associated with the differential. The differential in an integral indicates with respect to which variable the function will be integrated