If hikers walk along rugged trails, they might use a topographical map showing how steep the trails change. A topographical map contains curved lines, which are contour lines. Each contour line corresponds to the points on the map that have equal elevation. A level curve of a function of two variables\(f(x,y)\) is completely analogous to a contour line on a topographical map.
In mathematical, the contour line is known as level curves. A level
curve of a function \(f(x,y)\) is the curve of points \((x,y)\) where
\(f(x,y)\) is some constant value. A level curve is simply a
cross section
of the graph of \(z=f(x,y)\) taken at a constant value, say \(z=c\). A
function has many level curves, as one obtains a different level curve for
each value of \(c\) in the range of \(f(x,y)\). We can plot the level
curves for different constants \(c\) together in a level curve plot, which
is sometimes called a contour plot or contour line.
Let's work on a example to explore the curve line. Returning the function \(f(x,y)=x^2+y^2\). For some constant c, the level curve f(x,y)=c is the graph of \(c=x^2+y^2\). As long as \(c > 0\), this graph is a circle, as one can rewrite the equation for the level curve as\[\frac{x^2}{c}+\frac{y^2}{c} = 1\] For example, if \(c=1\), the level curve is the graph of \(x^2+y^2=1\). In the level curve plot of \(f(x,y)\) shown below, the smallest ellipse in the center is when \(c=1\). Working outward, the level curves are for \(c=2,3,\dots,6\) and bounded by \(R=[-2,2]\times[-2,2]\).
If we show this function in 3-Dimension, we will have an elliptic
paraboloid, and the level curve will vary across this elliptic paraboloid.
We can see the 3 D model in the following graph.
We can define the level curve:
If \(f(x,y)\) is a function of two variables \(x\) and \(y\), then the curve in an \((x,y)\)-coordinate system with points such that the \(x-\) and \(y-\) coordinate satisfy the equation\(f(x,y)=c\) is called the level curve with a (function) value equal to \(c\).
Notice that:
1.Level curves of a function of two variables can be drawn in an \((x,y)\)-coordinate system; the graph itself is drawn in an \((x,y,z)\)-coordinate system.
2. Level curves of the same function with different values cannot intersect.
Fig 1.Staff, POB Editorial. “A Construction Stakeout at Devils Tower National Monument.” Point of Beginning RSS, Point of Beginning, 12 Sept. 2020, https://www.pobonline.com/articles/102177-a-construction-stakeout-at-devils-tower-national-monument.
Fig 2. “Geology and Physiography of Devils Tower.” Pine Ridge, October 7, 2021. https://serc.carleton.edu/research_education/nativelands/pineridge/geology3.html. Staff, POB Editorial. “A Construction Stakeout at Devils Tower National Monument.” Point of Beginning RSS. Point of Beginning, September 12, 2020. https://www.pobonline.com/articles/102177-a-construction-stakeout-at-devils-tower-national-monument.
Author: Yueqi Li (Nicole)
Let's work on a example to explore the curve line. Returning the function \(f(x,y)=x^2+y^2\). For some constant c, the level curve f(x,y)=c is the graph of \(c=x^2+y^2\). As long as \(c > 0\), this graph is a circle, as one can rewrite the equation for the level curve as\[\frac{x^2}{c}+\frac{y^2}{c} = 1\] For example, if \(c=1\), the level curve is the graph of \(x^2+y^2=1\). In the level curve plot of \(f(x,y)\) shown below, the smallest ellipse in the center is when \(c=1\). Working outward, the level curves are for \(c=2,3,\dots,6\) and bounded by \(R=[-2,2]\times[-2,2]\).
If \(f(x,y)\) is a function of two variables \(x\) and \(y\), then the curve in an \((x,y)\)-coordinate system with points such that the \(x-\) and \(y-\) coordinate satisfy the equation\(f(x,y)=c\) is called the level curve with a (function) value equal to \(c\).
Notice that:
1.Level curves of a function of two variables can be drawn in an \((x,y)\)-coordinate system; the graph itself is drawn in an \((x,y,z)\)-coordinate system.
2. Level curves of the same function with different values cannot intersect.
Fig 1.Staff, POB Editorial. “A Construction Stakeout at Devils Tower National Monument.” Point of Beginning RSS, Point of Beginning, 12 Sept. 2020, https://www.pobonline.com/articles/102177-a-construction-stakeout-at-devils-tower-national-monument.
Fig 2. “Geology and Physiography of Devils Tower.” Pine Ridge, October 7, 2021. https://serc.carleton.edu/research_education/nativelands/pineridge/geology3.html. Staff, POB Editorial. “A Construction Stakeout at Devils Tower National Monument.” Point of Beginning RSS. Point of Beginning, September 12, 2020. https://www.pobonline.com/articles/102177-a-construction-stakeout-at-devils-tower-national-monument.
Author: Yueqi Li (Nicole)
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