Multivariable Calculus: Parameterize the curve of intersection

In this exercise, we parameterize the curve of intersection between the plane z=2x+2 and the paraboloid z=x^2+y^2-1. The curve is a tilted circle. By combining the equations of the two surfaces, we find a parametric description for the curve: r(t) = ( 1 + 2cos(t) , 2sin(t) , 4 + 4cos(t) ). The paramete t ranges from 0 to 2pi, allowing us to trace the entire curve of intersection.
#mathematics #math #multivariablecalculus #vectorcalculus #iitjammathematics #calculus3

[@mfn1311]

Nice video. In general parametrization is pretty easy, but needed a little refresher after not having looked at it for some weeks. Thanks.

[@bevinmaultsby]

Always good to practice!

[@armaansengupta]

Very nice video, I had a question regarding why we are allowed to just set x and y to those trigonometric functions. Is there a video or topic name you recommend to look up to see why it is we are allowed to do this?

[@bevinmaultsby]

To parametrize the curve of intersection, we want to find a coordinate description that traces out the curve in terms of a single variable t. Once we realize that the curve is a tilted circle, we can use the fact that any point on a circle of radius r centered at (h,k) can be parametrized as x=h+rcos⁡(t) and y=k+rsin⁡(t), where t will go from 0 to 2pi to take us all the way around the circle. (Trigonometric functions naturally describe circular motion in this way, so yes, we allowed to just do this!) I have some more examples of parametrization here: kzbin.info/www/bejne/pJa5iHSfpb-EsNk

[@filedotzip]

Get a better audio setup, or tune what you have, otherwise youre very professional and well paced

[@bevinmaultsby]

I’m still learning!