How Do You Parametrize the Curve of Intersection for Complex Surfaces?

[Karnage1993]

Homework Statement

Evaluate ##\int_{\gamma} F \ ds## where ##\gamma## is a parametrization of the curve of intersection of the surface ##z = x^4 + y^6## with the ellipsoid ##x^2 + 4y^2 + 9z^2 = 36## oriented in the counterclockwise direction when viewed from above.

Homework Equations

The Attempt at a Solution

The first thing I tried to do is plug the equation for the surface into the ellipsoid to get ##x^2 + 4y^2 + 9(x^4 + y^6)^2 = 36## which expands to ##9 x^8 + 18 x^4 y^6 + x^2 + 9 y^{12} + 4 y^2 = 36##. At this point, would I try to find the equation in terms of ##y##? Parametrizing this is where I seem to be stuck at.

 

[mighty2000]

Yes, you are on the right track. To parametrize the curve of intersection, you can use the equations of the two surfaces and solve for one variable in terms of the other two. In this case, you can solve for ##x## and ##y## in terms of ##z## using the equations ##z = x^4 + y^6## and ##x^2 + 4y^2 + 9z^2 = 36##. This will give you a parametrization of the curve in terms of ##z##.

Once you have the parametrization, you can use the formula ##\int_{\gamma} F \ ds = \int_{a}^{b} F(\gamma(t)) \cdot \gamma'(t) \ dt## where ##\gamma(t)## is the parametrization, ##a## and ##b## are the start and end points of the curve, and ##F## is the vector field.

In this case, the vector field ##F## will be ##F(x,y,z) = (x^2, y^2, z^2)##. You can plug in the parametrization for ##x##, ##y## and ##z## and evaluate the integral. Remember to also take into account the orientation of the curve when evaluating the integral.