![]() For aįunction of three variables, a level set is a surface in Set was a curve in two dimensions that we called a level curve. For a function of two variables, above, we saw that a level Instead, we can look at the level sets where the function isĬonstant. Most, you may have a hard time visualizing what this graph in four However, unless your mind is better at abstract visualization than To become comfortable with the idea of four dimensions. The graph is the set of points $(x,y,z,f(x,y,z))$. We would need four dimensions to draw its graph. Level surfacesįor a scalar-valued functions of three variables, $f : \R^3 \to \R$, Not surprisingly, the steepest slopes seem to be very near the summits of Eagle Mountain and Moose Mountain. The closer the contour lines are together, the steeper the slope of the land. Topographic maps are simply level curves of the elevation of the land (or water).įor instance, on standard United States Geological Survey maps, each contour line represents 10 feet of elevation above sea level,Īs in this topographic map of Eagle Mountain, the highest mountain (well. You can also change $c$ by dragging the red level curve. You can change $c$ by dragging the plane slicing the graph up or down with the mouse. The level curve $f(x,y)=c$ is shown in red in the level curve plot, which is the same as the slice of the graph $z=f(x,y)$ by the plane $z=c$. The graph of the function $f(x,y)=-x^2-2y^2$ is shown is the first panel along with a level curve plot in the second panel. ![]() Level curves of an elliptic paraboloid shown with graph. This slice is the intersection of the graph with the plane $z=c$. The key point is that a level curve $f(x,y)=c$ can be thought of as a horizontal slice of the graph at height $z=c$. The below graph illustrates the relationship between the level curves and the graph of the function. In the level curve plot of $f(x,y)$ shown below, the smallest ellipse (If $c$ is negative, then both denominators are positive.) ForĮxample, if $c=-1$, the level curve is the graph of $x^2 + 2y^2=1$. Rewrite the equation for the level curve as As long as $c<0$, this graph is an ellipse, as one can For someĬonstant $c$, the level curve $f(x,y)=c$ is the graph of We return to the above example function $f(x,y) = -x^2-2y^2$. Together in a level curve plot, which is sometimes called a contour We can plot the level curves for a bunch of different constants $c$ A function has many level curves, as one obtains aĭifferent level curve for each value of $c$ in the range of $f(x,y)$. A level curve is simply aĬross section of the graph of $z=f(x,y)$ taken at a constant value, ![]() Of two variables into a two-dimensional plot is through level curves.Ī level curve of a function $f(x,y)$ is the curve of points $(x,y)$ One way to collapse the graph of a scalar-valued function The nice part of of level sets is that they live in the same dimensions as the domain of the function.Ī level set of a function of two variables $f(x,y)$ is a curve in the two-dimensional $xy$-plane, called a level curve.Ī level set of a function of three variables $f(x,y,z)$ is a surface in three-dimensional space, called a level surface. Moreover, the graph of a function $f(x,y,z)$ of three variables would be the set of points $(x,y,z,f(x,y,z))$ in four dimensions,Īnd it would be difficult to imagine what such a graph would look like.Īnother way of visualizing a function is through level sets, i.e., the set of points in the domain of a function where the function is constant. To draw and visualize than two-dimensional plots. Three-dimensional plots, such as the above figure, are more difficult A graph of the function $f(x,y)=-x^2-2y^2$ over the domain $-2 \le x \le 2$ and $-2 \le y \le 2$.
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