PlanetMath: level curve

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (198)
Orphanage
Unclass'd
Unproven (498)
Corrections (24)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

level curve

(Definition)

The level curves (in German Niveaukurve, in French ligne de niveau) of a surface

$\displaystyle z = f(x,\,y)$

(1)

in $\mathbb{R}^3$ are the intersection curves of the surface and the planes $z = \,\mathrm{constant}$ . Thus the projections of the level curves on the

-plane have equations of the form

$\displaystyle f(x,\,y) = c$

(2)

where

is a constant.

For example, the level curves of the hyperbolic paraboloid are the rectangular hyperbolas (cf. this entry).

The gradient $f'_x(x,\,y)\,\vec{i}\!+\!f'_y(x,\,y)\,\vec{j}$ of the function in any point of the surface (1) is perpendicular to the level curve (2), since the slope of the gradient is $\displaystyle\frac{f'_y}{f'_x}$ and the slope of the level curve is $\displaystyle-\frac{f'_x}{f'_y}$ , whence the slopes are opposite inverses.

Analogically one can define the level surfaces (or contour surfaces)

$\displaystyle F(x,\,y,\,z) = c$

(3)

for a function

of three variables

. The gradient of

in a point $(x,\,y,\,z)$ is parallel to the surface normal of the level surface passing through this point.

"level curve" is owned by pahio.

(view preamble)

See Also: level set, convex angle

Other names:

contour curve, isopleth

Also defines:

level surface, contour surface

Keywords:

level set

This object's parent.

Attachments:: tilt curve (Definition) by pahio

Log in to rate this entry.
(view current ratings)

Cross-references: passing through, surface normal, parallel, variables, opposite inverses, slope, perpendicular, point, function, gradient, rectangular hyperbolas, equations, projections, planes, curves, intersection, surface
There are 3 references to this entry.

This is version 9 of level curve, born on 2007-10-17, modified 2008-06-11.
Object id is 10003, canonical name is ContourLine.
Accessed 1547 times total.

Classification:

AMS MSC:	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
	53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact