PlanetMath: isocline

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isocline

(Definition)

Let $\Gamma$ be a family of plane curves. The isocline of $\Gamma$ is the locus of the points, in which all members of $\Gamma$ have an equal slope.

If the family $\Gamma$ has the differential equation

$\displaystyle F(x,\,y,\,\frac{dy}{dx}) = 0,$

then the equation of any isocline of $\Gamma$ has the form

$\displaystyle F(x,\,y,\,K) = 0$

where

is constant.

For example, the family

$\displaystyle y = e^{Cx}$

of exponential curves satisfies the differential equation $\frac{dy}{dx} = Ce^{Cx}$ or $\frac{dy}{dx} = Cy$ , whence the isoclines are

, i.e. they are horizontal lines.

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"isocline" is owned by pahio.

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See Also: orthogonal curves

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Cross-references: lines, curves, equation, differential equation, slope, points, locus, plane curves
There are 2 references to this entry.

This is version 3 of isocline, born on 2008-05-31, modified 2008-05-31.
Object id is 10638, canonical name is Isocline.
Accessed 367 times total.

Classification:

AMS MSC:	51N05 (Geometry :: Analytic and descriptive geometry :: Descriptive geometry)
	53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
	53A25 (Differential geometry :: Classical differential geometry :: Differential line geometry)

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