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Complex Variables
Published in William S. Levine, Control System Fundamentals, 2019
Every analytic function w = f(z) can be viewed as a mapping from the z plane to the w plane. Suppose γ(t) is a differentiable curve passing through a point z0 at time t = 0. The curve Γ = f(y) is a curve passing through the point w0 = f(z0). An application of the chain rule gives () Γ′(0)=f′(z0)γ′(0).
Applications of the Cauchy Theory
Published in Steven G. Krantz, Complex Variables, 2019
Suppose that the polynomial p has a simple zero at z0 and let γ be a simple closed, continuously differentiable curve that encircles z0 (oriented in the counterclockwise direction). What can you say about the value of 12πi∮γp′(ζ)p(ζ)dζ?
A Review of Calculus
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
If the differentiable curve is given parametrically by a set of points (x, y) where x = x(t), y = y(t), a ≤ t ≤ b, are each differentiable functions of t, then its length is given by () ∫abx′(t)2+y′(t)2dt
A global-local approach for dynamic stress evaluation of lazy wave flexible risers subjected to random wave and vessel motion
Published in Ships and Offshore Structures, 2023
Weidong Ruan, Qinlin Nie, Yutian Lu, Miaoyi Chen, Dahui Liu, Bo Sun
Darboux frame is adopted to describe the geometrical features of the helical strips wrapped around the cylinder. As depicted in Figure 11, the Darboux frame is built by three orthogonal vectors (G1, G2, G3), where G1, G2 and G3 are the vector tangent to the helical strip, vector normal to the contact surface and vector tangent to the contact surface. In this Darboux frame, a differentiable curve of the helical strip in three-dimensional space can be defined as: where κ1 denotes the torsion curvature of the helical strip; κ2 and κ3 are the geodesic curvature and normal curvature of the helical strip, respectively; S is the surface equation of cylindrical surface after bending deformation.
A non-Secant quasi-Newton Method for Unconstrained Nonlinear Optimization
Published in Cogent Engineering, 2022
We will use data from three most recent iterations to derive the new Secant-like relationship that the updated Hessian (or its inverse) must fulfil, just as we did with the nonlinear multi-step quasi-Newton methods (equations (5–10)) discussed earlier, thus choosing in (6). In particular, an interpolation of the points , and is constructed using a differentiable curve in , , such that , and .
Tseng's extragradient algorithm for pseudomonotone variational inequalities on Hadamard manifolds
Published in Applicable Analysis, 2022
Jingjing Fan, Xiaolong Qin, Bing Tan
Let be a finite dimensional differentiable manifold. The set of all tangents at is called a tangent space of at , which forms a vector space of the same dimension as and is denoted by . The tangent bundle of is denoted by , which is naturally a manifold. We denote by the scalar product on with the associated norm , where the subscript x is sometimes omitted. A differentiable manifold with a Riemannian metric is called a Riemannian manifold. Let be a piecewise differentiable curve joining to in , we can define the length of . The minimal length of all such curves joining x to y is called the Riemannian distance and it is denoted by .