Lesson 83: Applications of Nonlinear Systems in 3D
Nonlinear systems with three variables appear frequently in physics, engineering, and economics. Solutions correspond to points where three surfaces intersect in 3D space.
Examples include spheres intersecting planes, paraboloids intersecting lines, and optimization problems in three dimensions.
Examples
Example 1 — Sphere and Plane:
Solve for intersection of the sphere x² + y² + z² = 14 and plane x + y + z = 6.
Substitute z = 6 − x − y → x² + y² + (6 − x − y)² = 14 → x² + y² + 36 − 12x − 12y + x² + y² + 2xy = 14 → 2x² + 2y² + 2xy − 12x − 12y + 22 = 0 → x² + y² + xy − 6x − 6y + 11 = 0 Solve for all real (x, y, z) solutions.
Example 2 — Optimization Problem:
Maximize volume V = xyz subject to constraint x² + y² + z² = 12. This forms a nonlinear system with three variables. Use substitution or Lagrange multipliers for solution.
Example 3 — Intersection of Cylinder and Plane:
Cylinder: x² + y² = 9
Plane: z = x + y
Intersection: x² + y² = 9, z = x + y → points on the cylinder mapped to plane.
Practice Problems
- Find the intersection of x² + y² + z² = 27 and x + y + z = 9.
- Determine points where z = x² + y² intersects x + 2y − z = 0.
- Solve the system: x² + y² = 16, y + z = 6, x − z = 2.
- Find all points where the sphere x² + y² + z² = 25 intersects the plane z = x + 3.
- Optimization: Maximize volume of a box inscribed in sphere x² + y² + z² = 36.