Find a vector function that represents the curve of intersection of the two surfaces:

the cylinder x^(2)+y^(2)=9 and the surface z=x^(2)-y^(2).

To find a vector function that represents the curve of intersection of the two surfaces, we need to parametrize both surfaces simultaneously. The two surfaces given are:

1. The cylinder: \( x^2 + y^2 = 9 \)
2. The surface: \( z = x^2 - y^2 \)

### Step 1: Parametrizing the Cylinder
The equation \( x^2 + y^2 = 9 \) suggests a circle in the \( xy \)-plane with radius 3. This can be parametrized using trigonometric functions:

\[
x = 3 \cos(t), \quad y = 3 \sin(t)
\]

where \( t \) is the parameter, and \( t \in [0, 2\pi] \) covers the full circle.

### Step 2: Using the Surface Equation for \( z \)
The surface equation is \( z = x^2 - y^2 \). Substituting the parametric expressions for \( x \) and \( y \) from the cylinder into this equation:

\[
z = (3 \cos(t))^2 - (3 \sin(t))^2 = 9 \cos^2(t) - 9 \sin^2(t)
\]

Using the trigonometric identity \( \cos(2t) = \cos^2(t) - \sin^2(t) \), this simplifies to:

\[
z = 9 (\cos^2(t) - \sin^2(t)) = 9 \cos(2t)
\]

### Step 3: Forming the Vector Function
Now, we have parametric expressions for \( x \), \( y \), and \( z \):

\[
x = 3 \cos(t), \quad y = 3 \sin(t), \quad z = 9 \cos(2t)
\]

Thus, the vector function that represents the curve of intersection is:

\[
\mathbf{r}(t) = \langle 3 \cos(t), 3 \sin(t), 9 \cos(2t) \rangle
\]

where \( t \in [0, 2\pi] \).

### Final Answer:
The vector function is:

\[
\mathbf{r}(t) = \langle 3 \cos(t), 3 \sin(t), 9 \cos(2t) \rangle
\]

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Would you like more details on any of the steps, or do you have any other questions?

Here are 5 related questions:
1. How would you parametrize a general cylinder \( x^2 + y^2 = r^2 \)?
2. What is the significance of trigonometric identities in parametrizing surfaces?
3. How do we parametrize a different surface, such as \( z = x + y \), intersecting a cylinder?
4. How would the vector function change if the cylinder had an elliptical cross-section?
5. Can the method of parametrization be applied to three surfaces?

**Tip:** Always look for symmetry in the surfaces to simplify the parametrization process, such as using trigonometric functions for circular shapes.

Find a vector function that represents the curve of intersection of the two surfaces: the cylinder x^(2)+y^(2)=9 and the surface z=x^(2)-y^(2).

Discover how to find the parametric vector function representing the curve of intersection of the cylinder x^2 + y^2 = 9 and the surface z = x^2 - y^2. This solution uses vector calculus and trigonometric identities to parametrize the curve. Ideal for university-level students studying advanced calculus.

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