Parabolic Cylinder — Definition, Formula & Examples

A parabolic cylinder is a three-dimensional surface created by taking a parabola in one plane and extending it infinitely along a direction perpendicular to that plane. Unlike a paraboloid, it has no curvature along one of its axes.

A parabolic cylinder is a quadric surface in $\mathbb{R}^3$ defined by an equation of the form $y = ax^2$ (or any cyclic permutation of variables), where one variable is entirely absent. The missing variable indicates the axis along which the parabolic cross-section is translated, making the surface a ruled surface generated by lines parallel to that axis.

Key Formula

z = ax^2

Where:

$a$ = A nonzero constant controlling how steeply the parabola opens
$x$ = Variable in the direction of parabolic curvature
$z$ = Height of the surface above the xy-plane
$y$ = Absent from the equation — the surface extends infinitely along the y-axis

How It Works

To identify a parabolic cylinder, look for a quadratic equation in three-dimensional space where one of the three variables (

x

y

, or

z

) does not appear. The two variables that do appear define a parabola in their coordinate plane, and the surface extends without bound along the axis of the missing variable. Every cross-section parallel to the parabola's plane is an identical copy of that parabola. Cross-sections perpendicular to the parabola's plane are straight lines, which is why the surface qualifies as a cylinder in the generalized geometric sense.

Worked Example

Problem: Describe the surface defined by z = 2x² in three-dimensional space and find the cross-section at y = 5.

Identify the surface type: The equation z = 2x² involves only x and z. Because y is missing, this is a parabolic cylinder extending along the y-axis.

z = 2x^2 \quad (\text{no } y \text{ term})

Describe cross-sections: Every cross-section at a fixed value of y is the same parabola z = 2x² in the xz-plane.

\text{At } y = 5: \quad z = 2x^2

Interpret geometrically: The cross-section at y = 5 is identical to the cross-section at y = 0 or any other y-value. The surface looks like a parabolic trough running parallel to the y-axis.

Answer: The surface is a parabolic cylinder (a trough shape) extending along the y-axis, and its cross-section at y = 5 is the parabola z = 2x².

Why It Matters

Parabolic cylinders appear in multivariable calculus when you set up double and triple integrals over regions bounded by quadratic surfaces. They also model physical structures like parabolic trough solar collectors, where the cylindrical shape focuses light along a line rather than at a single point.

Common Mistakes

Mistake: Confusing a parabolic cylinder with a paraboloid because both involve squared terms.

Correction: A paraboloid (e.g., z = x² + y²) curves in two directions, while a parabolic cylinder (e.g., z = x²) curves in only one direction and extends straight along the missing variable's axis.