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The slope field can be defined for the following type of differential equations ′ = (,), 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.
Three integral curves for the slope field corresponding to the differential equation dy / dx = x 2 − x − 2. If the differential equation is represented as a vector field or slope field , then the corresponding integral curves are tangent to the field at each point.
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.
Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis , functional analysis , differential geometry , measure theory , and abstract algebra .
Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated. The idea is that while the curve is initially unknown, its starting point, which we denote by , is known (see Figure 1). Then, from the ...
The flows in the vector field indicate the time-evolution of the system the differential equation describes. In this way, phase planes are useful in visualizing the behaviour of physical systems; in particular, of oscillatory systems such as predator-prey models (see Lotka–Volterra equations). In these models the phase paths can "spiral in ...
Because the slope of the orthogonal trajectory at a point (,) is the negative multiplicative inverse of the slope of the given curve at this point, the orthogonal trajectory satisfies the differential equation of first order
Discrete calculus is used for modeling either directly or indirectly as a discretization of infinitesimal calculus in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled. It allows one ...