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The slope field of =, with the blue, red, and turquoise lines being +, , and , respectively.. A slope field (also called a direction field [1]) is a graphical representation of the solutions to a first-order differential equation [2] of a scalar function.
Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of y' = xy. The solution curves are y = C e x 2 / 2 {\displaystyle y=Ce^{x^{2}/2}} . Given a family of curves , assumed to be differentiable , an isocline for that family is formed by the set of points at which some member of the family attains a given slope .
If we draw a graph of the logistic map (), we can observe that the slope of the tangent at the fixed point exceeds 1 at the boundary = and becomes unstable. At the same time, two new intersections appear, which are the periodic points x f 1 ( 2 ) {\displaystyle x_{f1}^{(2)}} and x f 2 ( 2 ) {\displaystyle x_{f2}^{(2)}} .
Keeping in mind that the slope is at most , the problem now presents itself as to whether the next point should be at (+,) or (+, +). Perhaps intuitively, the point should be chosen based upon which is closer to the line at +. If it is closer to the former then include the former point on the line, if the latter then the latter.
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .
World No. 1 Jannik Sinner fought through blistering heat and an undisclosed ailment to defeat Holger Rune 6-3, 3-6, 6-3, 6-2 and advance to the quarterfinals of the Australian Open on Monday.
Once a DC operating point is defined by the DC load line, an AC load line can be drawn through the Q point. The AC load line is a straight line with a slope equal to the AC impedance facing the nonlinear device, which is in general different from the DC resistance. The ratio of AC voltage to current in the device is defined by this line.
From ancient history to the modern day, the clitoris has been discredited, dismissed and deleted -- and women's pleasure has often been left out of the conversation entirely. Now, an underground art movement led by artist Sophia Wallace is emerging across the globe to challenge the lies, question the myths and rewrite the rules around sex and the female body.