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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.
The word comes from the Greek words ἴσος (isos), meaning "same", and the κλίνειν (klenein), meaning "make to slope". Generally, an isocline will itself have the shape of a curve or the union of a small number of curves. Isoclines are often used as a graphical method of solving ordinary differential equations.
The second-order autonomous equation = (, ′) is more difficult, but it can be solved [2] by introducing the new variable = and expressing the second derivative of via the chain rule as = = = so that the original equation becomes = (,) which is a first order equation containing no reference to the independent variable .
Polynomial curves fitting points generated with a sine function. The black dotted line is the "true" data, the red line is a first degree polynomial, the green line is second degree, the orange line is third degree and the blue line is fourth degree. The first degree polynomial equation = + is a line with slope a. A line will connect any two ...
where is the slope and is the y-intercept. Because this is a function of only , it can't represent a vertical line. Therefore, it would be useful to make this equation written as a function of both and, to be able to draw
The next step is to multiply the above value by the step size , which we take equal to one here: h ⋅ f ( y 0 ) = 1 ⋅ 1 = 1. {\displaystyle h\cdot f(y_{0})=1\cdot 1=1.} Since the step size is the change in t {\displaystyle t} , when we multiply the step size and the slope of the tangent, we get a change in y {\displaystyle y} value.
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 ...
In the computer science subfields of computer-aided design and computer graphics, the term spline more frequently refers to a piecewise polynomial curve. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through ...