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Parametrized Surfaces Just as you describe curves in the plane parametrically with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section 16.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.$$\begin{array}{l}x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2 \\0 \leq v \leq 2 \pi\end{array}$$

$$0 \leq u \leq 2$$

Calculus 3

Chapter 14

Partial Derivatives

Section 1

Functions of Several Variables

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12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Parametrized Surfaces Just…

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Just as you describe curve…

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Okay. What we want to go ahead and do is we want to graph the Parametric equations. Ah x equal to you Times co sign of E. Why equal to you? Time sign of e and C is equal to V and we want you to be between zero and two and V to be lips between, um zero and two pi inclusive. And not only do we want to grab the Parametric thes parametric equations, we also want to in the X Y plane two graf several level curves. Joe, let's go ahead and switch to a three D graph for, um to go ahead and graff the Parametric equations. Um, I'm happened. Thio use a free online three D calculator through Geo Jabra And in order to do, um, Parametric equations and three dimensional Sze, we have to use a built in function called surface and you put in your ex your y, your z and then your parameters You going from 0 to 2 and then, um v from 0 to 2 pi. And so here is the graph, um, of those parametric equations and I can actually rotate the, um, blue accesses e read his ex and green is why. And so if I want to know what what, um, what that will look like. Um, if I looked down on the X y plane, I noticed that I just see circles, even though this does not certain look circular. Um, when I looked down upon it, I do on the contours to look circular. So I know that's what I'm kind of going for in the in the two dimensional X y plane. And so let's go ahead and go back and develop, um, the Cartesian equations for X and y. And so I know we're going towards circles, so we know that X is equal to you. Co sign v Wise Eagle, you sign V and so we're gonna square. So we're going back to Rectangular. So X squared plus y squared is equal. To use squared times co sign squared V plus sine squared V, which, of course, is one so X squared. Plus y squared is equal to use squared and so let's go ahead and graft several level curves. So let's do X squared. Plus y squared is equal to 1/4 X squared plus y squared is equal to one and X squared plus y squared is equal 24 So let's go ahead and do those three. So we're gonna go to a two dimensional graph e and i'ma put in X squared. Plus why squared equal to 1/4? That's a circle with a radius of 1/2 and then X beard plus y squared equal to one and ex wits x squared. This why squared equal before? And so there are, um, several level curves and the X Y plane.

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