level curve
of a function of two variables are all of the curves with the equations f(x,y)=k where k is some constant in the range of f (topographical map)
A level curve, also known as a contour line, is a curve on a two-dimensional surface that represents points on the surface at a constant value of some parameter or function. For example, on a topographic map, the level curves represent points of equal elevation. In mathematics, level curves are often used to visualize and study functions of two variables.
To construct a level curve for a function of two variables, we first choose a value for the function. Then, we find all the points on the surface where the function takes that value, and we plot them on a two-dimensional plane. The resulting curve is a level curve.
Level curves are useful for several reasons. They can help us understand the behavior of a function and identify important features such as maxima, minima, and saddle points. They can also be used to find paths of constant value or to visualize changes in the function over time. In addition, level curves are often used in applications such as engineering, physics, and economics to represent phenomena that vary over a two-dimensional surface.
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