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Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have $ u $ constant and which have $ v $ constant.

$ \textbf{r}(u, v) = \langle u, v^3, -v \rangle $,

$ -2 \leqslant u \leqslant 2 $, $ -2 \leqslant v \leqslant 2 $

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Vector Calculus

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Campbell University

Harvey Mudd College

Baylor University

Boston College

So with this problem, we have a three D graph of the Parametric surface. Onda, we want to see that there are two types of grid curves. So we'll show you the picture right here. So it looks something like this Almost like a sled shape. Um, like this and then we have curved grid lines like this, But then we also have, um, straight grid lines such as thes ones right here. And what we see is that when we have the Parametric station of the surface, which is our of UV equal to you, the cube Negative V, we see that if we substitute V equals zero, then we just get you 00 And that represents a straight perpendicular line to the Y Z plane. So v being constant are these lines right here? So that's V constant. And then if we make you constant, then we're gonna end up getting the grid curves that are not straight lines, which are these ones right here. And that will be our final answer for the problem. We could also look at the projection of the surface. Thes projections would look more just like straight grids, but more importantly, we wanna look at this and recognize when we have you constant and where we have the constant

California Baptist University

Vector Calculus