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The slope field can be defined for the following type of differential equations y ′ = f ( x , y ) , {\displaystyle y'=f(x,y),} which can be interpreted geometrically as giving the slope of the tangent to the graph of the differential equation's solution ( integral curve ) at each point ( x , y ) as a function of the point coordinates.
Two-dimensional slice through 3D Perlin noise at z = 0. Perlin noise is a type of gradient noise developed by Ken Perlin in 1983. It has many uses, including but not limited to: procedurally generating terrain, applying pseudo-random changes to a variable, and assisting in the creation of image textures.
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 .
l = slope length α = angle of inclination. The grade (US) or gradient (UK) (also called stepth, slope, incline, mainfall, pitch or rise) of a physical feature, landform or constructed line is either the elevation angle of that surface to the horizontal or its tangent. It is a special case of the slope, where zero indicates horizontality. A ...
It uses a combination of the energy, momentum, and continuity equations to determine water depth with a given a friction slope (), channel slope (), channel geometry, and also a given flow rate. In practice, this technique is widely used through the computer program HEC-RAS , developed by the US Army Corps of Engineers Hydrologic Engineering ...
Many of the preceding results remain valid when the field of definition of E is a number field K, that is to say, a finite field extension of Q. In particular, the group E(K) of K -rational points of an elliptic curve E defined over K is finitely generated, which generalizes the Mordell–Weil theorem above.
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems .
The flow field around an airplane is a vector field in R 3, here visualized by bubbles that follow the streamlines showing a wingtip vortex. Vector fields are commonly used to create patterns in computer graphics. Here: abstract composition of curves following a vector field generated with OpenSimplex noise.