Thread: Calculus III: Vector Functions and Space Curves

1. Calculus III: Vector Functions and Space Curves

If someone could please help me with this problem it would be most appreciated! I don't even know where to start!

Find a vector function that represents the curve of intersection of the two surfaces (in the 3D space):

The paraboloid: $\displaystyle z = 4x^2 + y^2$
The parabolic cylinder: $\displaystyle y = x^2$

2. Originally Posted by saxyliz
If someone could please help me with this problem it would be most appreciated! I don't even know where to start!

Find a vector function that represents the curve of intersection of the two surfaces (in the 3D space):

The paraboloid: $\displaystyle z = 4x^2 + y^2$
The parabolic cylinder: $\displaystyle y = x^2$
plug in $\displaystyle y = x^2$ into $\displaystyle z$, so that $\displaystyle z = 4x^2 + x^4$

now parametrize the curve using $\displaystyle x = t$ as the parameter

3. What exactly do to mean by "parametrize the curve?"

Is it:
$\displaystyle z = 4t^2 + t^4$
?

4. Originally Posted by saxyliz
What exactly do to mean by "parametrize the curve?"

Is it:
$\displaystyle z = 4t^2 + t^4$
?
the vector function is $\displaystyle r(t) = \left< x(t), y(t), z(t) \right> = \left< t, t^2 , 4t^2 + t^4 \right>$

5. Oh! I get it now! Thanks a lot!