Level Curves and Surfaces A level curve is a curve that lies entirely on a level surface. In other words, it is a curve whose points are all at the same...
Level Curves and Surfaces
A level curve is a curve that lies entirely on a level surface. In other words, it is a curve whose points are all at the same height. For example, the curve y = x^2 is a level curve, since it lies on the surface of the planet Earth at the same height above sea level for all points on the curve.
A level surface is a surface that is defined by a set of equations that are equal to a constant. In other words, it is a surface that is composed of points that have the same value of a certain function. For example, the surface z = x^2 + y^2 + z^3 is a level surface, since it is defined by the equation z = x^2 + y^2 + z^3.
The relationship between level curves and surfaces is one of duality. A level curve is a curve that is tangent to a surface, while a surface can be represented by a level curve. For example, the curve y = x^2 is a level curve, while the surface z = x^2 + y^2 is a surface.
Examples:
The curve y = x^2 is a level curve, since it lies on the surface of the Earth at the same height above sea level for all points on the curve.
The surface z = x^2 + y^2 + z^3 is a level surface, since it is defined by the equation z = x^2 + y^2 + z^3.
The curve y = x^2 - y^2 is a level curve, since it lies on the surface of the Earth at the same height above sea level for all points on the curve.
The surface z = x^2 + y^2 - z is a level surface, since it is defined by the equation z = x^2 + y^2 - z