Level surface of a function of three variables

Objective:  Find the level surface of a function of three variables.

In a function of two variables, we have level curves which are defined in the xy plane. In a function of three variables, level curves are called surfaces

Definition

Level surfaces in a function of three variables

There are several types of level surfaces depending on the form of the function of three variables. Let's limit ourselves to three types.: planes, quadric surfaces and spheres.  

Plane

When the function f(x,y,z) is a linear function leading to the equation  ax + by + cz = d where a,b, c and are contants, the level surface is a plane. Planes are flat, have no curvature and extend infinitely.

Quadric surfaces

If the function f(x,y,z) is a quadric function leading to the equation ax² + by² + cz² = d which is a second degree polynomial, the level surface is a quadric surface. Quadric surfaces can be ellipsoid, hyperboloid, paraboloid, cone and cylinder depending on the form of the quadric equation.

Ellipsoid

Hyperboloid

There are two types:

1.Hyperboloid of one sheet

 

2. Hyperboloid of 2 sheets

Consists of two separate surfaces like two bowls facing each other.

Equation:

Paraboloid

There are two types:

 

Cone

Sphere

Example

Solution

Practice

Find the level surface of the function of 3 variables and describe it:

a) 

 

b)