find parametric equations of the curve given by the intersection of the surfaces 
The paraboloid: z=3x^2+y^2
The parabolic cylinder y=x^2

To find the parametric equations of the curve given by the intersection of the paraboloid \( z = 3x^2 + y^2 \) and the parabolic cylinder \( y = x^2 \), we can proceed as follows:

### Step 1: Substitute \( y = x^2 \) into the equation of the paraboloid.

The equation of the paraboloid is:
\[
z = 3x^2 + y^2
\]
Since \( y = x^2 \), substitute this into the equation of the paraboloid:
\[
z = 3x^2 + (x^2)^2 = 3x^2 + x^4
\]
Thus, the equation for \( z \) in terms of \( x \) becomes:
\[
z = x^4 + 3x^2
\]

### Step 2: Parametrize \( x \) as a parameter.

Let \( x = t \), where \( t \) is the parameter. Using this, we can parametrize the other variables:

- \( x = t \)
- \( y = t^2 \) (from the equation of the cylinder \( y = x^2 \))
- \( z = t^4 + 3t^2 \) (from the modified equation of the paraboloid)

### Step 3: Write the parametric equations.

The parametric equations for the curve of intersection are:
\[
x(t) = t
\]
\[
y(t) = t^2
\]
\[
z(t) = t^4 + 3t^2
\]

Thus, the parametric equations of the curve are:
\[
(x(t), y(t), z(t)) = (t, t^2, t^4 + 3t^2)
\]

Would you like more details on how to visualize this curve or the surfaces?

Here are 5 relative questions to deepen your understanding:
1. How would the parametric equations change if the cylinder was \( y = x^3 \) instead of \( y = x^2 \)?
2. What is the geometric interpretation of the curve formed by this intersection?
3. How can you find the tangent vector to the curve at any point?
4. How would the parametric equations change if the paraboloid had the form \( z = 4x^2 + y^2 \)?
5. What happens if you shift the paraboloid or cylinder by translating their equations?

**Tip:** Parametric equations offer a flexible way to describe curves in space and are especially useful for understanding intersections of complex surfaces.

find parametric equations of the curve given by the intersection of the surfaces: The paraboloid z=3x^2+y^2 and the parabolic cylinder y=x^2

Learn how to find the parametric equations of the curve formed by the intersection of the paraboloid z = 3x^2 + y^2 and the parabolic cylinder y = x^2. This detailed solution explores parametric equations, surface intersections, and provides step-by-step calculations. Suitable for undergraduate students studying Calculus III.

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