Identify the surface with the given vector equation. $\mathbf{r}(s, t) = \langle s, t, t^2 - s^2 \rangle$

• plane
• circular paraboloid
• hyperbolic paraboloid
• circular cylinder
• elliptic cylinder

Neetesh Kumar | November 14, 2024

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This is the solution to Math 1D
Assignment: 16.6 Question Number 5
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Step-by-step solution:

To determine the surface type, let’s analyze the components of the given vector equation: $\mathbf{r}(s, t) = \langle s, t, t^2 - s^2 \rangle$

Breaking down into coordinates:

• For the x-coordinate: $x = s$
• For the y-coordinate: $y = t$
• For the z-coordinate: $z = t^2 - s^2$

We can rewrite the $z$-coordinate equation in terms of $x$ and $y$: $z = y^2 - x^2$

Step 1: Recognize the Surface Equation

The equation $z = y^2 - x^2$ represents a hyperbolic paraboloid. This is a classic form of a hyperbolic paraboloid, where the cross-sections in the $xz$-plane and $yz$-plane are parabolas, and in the $xy$-plane, the equation represents hyperbolic traces.

Step 2: Verify with the Options

The equation $z = y^2 - x^2$ confirms that the surface is a hyperbolic paraboloid, as this is the standard form for such surfaces in three-dimensional space.

Answer:

The correct option is C: hyperbolic paraboloid

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