Describe the level surfaces of
g(x; y; z) = x2 - 2y2 + z2
To understand the level surfaces of the function g(x, y, z) = x^2 - 2y^2 + z^2, we need to examine what level surfaces mean in the context of a function.
A level surface of a function represents a set of points in three-dimensional space where the function has a constant value. In other words, it is a surface where the function remains constant.
In this case, the function g(x, y, z) is expressed in terms of three variables x, y, and z. The level surfaces of this function will consist of all the points in space where g(x, y, z) is equal to a constant value.
Let's consider different constant values for g(x, y, z) and examine the resulting level surfaces:
1. g(x, y, z) = 0:
When g(x, y, z) is set to zero, we have the equation x^2 - 2y^2 + z^2 = 0. This represents the origin (0, 0, 0) since all variables are being squared and added together. Therefore, the level surface is a single point at the origin.
2. g(x, y, z) = 1:
When g(x, y, z) is set to one, we have the equation x^2 - 2y^2 + z^2 = 1. This equation represents an elliptical shape centered at the origin. The surface expands outward from the center as the value of g increases.
3. g(x, y, z) = -1:
When g(x, y, z) is set to negative one, we have the equation x^2 - 2y^2 + z^2 = -1. This equation does not have any real solutions since the sum of squares cannot be negative for real numbers. Therefore, this level surface is empty.
4. g(x, y, z) = k, where k is a positive constant:
When g(x, y, z) is set to a positive constant k, we have the equation x^2 - 2y^2 + z^2 = k. This equation represents a family of nested ellipsoidal surfaces centered at the origin. As k increases, the ellipsoids become larger.
Overall, the level surfaces of the function g(x, y, z) = x^2 - 2y^2 + z^2 consist of points, ellipsoids, and empty spaces depending on the chosen constant value.