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Monitoring and Diagnostics
Published in S.V. Kulkarni, S.A. Khaparde, Transformer Engineering, 2017
The other two partial derivatives with respect to the x and y variables can be similarly expressed. The partial derivative with respect to time is written as () ∂fn(i,j,k)∂t=fn+1/2(i,j,k)−fn−1/2(i,j,k)Δt.
Mapping Boolean Expressions
Published in Eugene D. Fabricius, Modern Digital Design and Switching Theory, 2017
In calculus, the partial derivative of a function Y(A,B,C,D) with respect to variable A can be written as () YA=∂Y∂A=limΔA→0=Y(A+ΔA,B,C,D)-Y(A,B,C,D)ΔA
Differentiating fields
Published in A.V. Durrant, Vectors in Physics and Engineering, 2019
The curly dee is used to distinguish the partial derivative from the ordinary derivative df/dx of f(x). Notice that the coordinates y and z do not change when a displacement iΔx is made, and so the symbols y and z in the definition (5.2) stay constant as Δx goes to zero (Δx → 0). The definition (5.2) therefore has the same form as the definition of the derivative df/dx (Eq (3.13)). It follows that the partial derivative ∂Φ/dx can be calculated using the familiar rules of differentiation for functions f (x) of a single variable x, provided we regard y and z in the expression for Δ as constants. For example, the partial derivative of the function
Optimal control for uncertain random continuous-time systems
Published in Optimization, 2023
Let be twice differentiable on . Then we have where is the partial derivative of function in t, and are the gradients of function in and , respectively, is the Hessian matrix of function in .
Pre-service teachers’ understanding of the derivative of a function at a point
Published in International Journal of Mathematical Education in Science and Technology, 2023
María Fernanda Vargas González, José Antonio Fernández-Plaza, Juan Francisco Ruiz Hidalgo
Based on the terms, conventions and notations used by pre-service teachers to delimit the conceptual field, their definitions could be grouped, through content analysis, under five categories: The first category –16 replies– includes replies that defined the derivative of a function at a point solely as the slope of the line tangent to the graph of a function at the point. Examples are listed below.The derivative is the slope of the function.The derivative is the value of the tangent.The derivative is the tangent point of the function.
Traffic flow estimation at error prone locations using dynamic traffic flow modeling
Published in Transportation Letters, 2019
Shrikant Fulari, Ajitha Thankappan, Lelitha Vanajakshi, Shankar Subramanian
Here, A is the matrix of the partial derivative of f with respect to x, W is the matrix of the partial derivative of f with respect to w, c is the vector of the partial derivative of g with respect to x and m is the partial derivative of g with respect to vn. Now, the following steps were followed recursively for estimation using EKF:The a priori estimate in the (k + 1)th interval of time was obtained throughThe a priori error covariance in the (k + 1)th interval of time was obtained throughThe Kalman gain kk+1 was calculated throughThen, the a posteriori state estimate was calculated throughFinally, the a posteriori error covariance was obtained through