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What is a level curve?
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What is a level curve of a function of two variables?
Definition. Given a function f (x, y) f (x, y) and a number c c in the range of f f, a level curve of a function of two variables for the value c c is defined to be the set of points satisfying the equation f (x, y) =c f (x, y) = c.
When the number of independent variables is two, a level set is called a level curve, also known as contour line or isoline; so a level curve is the set of all real-valued solutions of an equation in two variables x1 and x2.
Level curves. One way to collapse the graph of a scalar-valued function of two variables into a two-dimensional plot is through level curves. A level curve of a function f(x, y) f (x, y) is the curve of points (x, y) (x, y) where f(x, y) f (x, y) is some constant value.
A level curve of a function f(x, y) is a cross section of a three-dimensional figure, projected onto an x-y plane. Level curves tell us something about heights on a landscape; they are used to create three-dimensional surfaces from two-dimensional contours.
Nov 16, 2022 · The level curves of the function \(z = f\left( {x,y} \right)\) are two dimensional curves we get by setting \(z = k\), where \(k\) is any number. So the equations of the level curves are \(f\left( {x,y} \right) = k\).
Recall that the level curves of a function \(f(x, y)\) are the curves given by \(f(x, y) =\) constant. Recall also that the gradient \(\nabla f\) is orthogonal to the level curves of \(f\)
A level curve is just a 2D plot of the curve f(x,y) = k, for some constant value k. Thus by plotting a series of these we can get a 2D picture of what the three-dimensional surface looks like.
