A parametrization has to be in rectangular coordinates (x,y,z), so we can't just describe the surface in cylindrical coordinates and say we're done. ![]() Many of our surfaces will be relatively easy to describe using cylindrical coordinates (r,θ,z), which have been covered in class. As in last week's lab, circular regions will be very important, and there is one technique in particular that you should learn. Let's look at more examples of how to parametrize surfaces. People using Mathematica 5.0 needn't worry about this. The command ParametricPlot will not accept more than one parameter! (So you can't give it ranges for, say, y and z.)Īlso, most of this lab uses ParametricPlot3D, so if you're using Mathematica version 4.x it would be a good idea to execute the command Off to get rid of those blue error messages about compiling functions. If you want to plot a surface, however, you must use ParametricPlot3D. Note: if a curve is in the xy-plane, you can use ParametricPlot, or you can use ParametricPlot3D by letting z(t)=0. Surfaces are two-dimensional creatures, and so they require two parameters. ![]() Curves are like a line, a one-dimensional object, so they require one parameter. Note that we're using two parameters now, which is different than when we plotted curves. Now we can plot the graph of f using ParametricPlot3D:Īnd you can see that this is a paraboloid opening in the direction of the positive x-axis, as desired. ![]() Define f(y,z)=( +, y, z):Įssentially we're saying that y and z are our parameters. Think back to last week when we did a "trivial parametrization" of the graph of y=g(x) by setting f(x) equal to (x, g(x)). How can we get a true graph of x = + from Mathematica? Using parametric equations is one possibility. Normally we give it a function of x and y, and since the z-axis represents height, this fits in to our view of the world. The reason is that Plot3D expects a function of two variables, and it interprets the values of the function as the height. We get exactly the same picture as before! The paraboloid opens upwards, which is incorrect. Suppose we tried to plot this with Plot3D: For example, what if we wanted to plot the graph of x = + ? You should know by looking at this equation that this is a paraboloid, just like our picture above, except this one opens in the direction of the positive x-axis. Mathematica can draw a great picture of graphs like this, but Plot3D has its limitations. Often our surfaces will be (a piece of) the graph of a function z=f(x,y), such as this example: The following pictures both show surfaces the last one is a closed surface, while the first is not. Sometimes a surface encloses a solid region in space. It's like a piece of paper, or a sheet of rubber, but it doesn't have to be flat it can be bent, curved, or even have holes in it. We haven't rigorously defined surfaces yet, but you probably have an intuitive idea of what a surface is. This week we'll do the same thing with surfaces. You also had the chance to find the parametrizations for various curves. Last week you learned how to graph curves using ParametricPlot and ParametricPlot3D. I don't know what's involved in terms of lists, but it seems to me this is fundamentally important, to respect accepted notation and not give quirky objections and insist on usually unreadable tangles of lists.Īfter all, this matrix layout evolved for both readability and intuitiveness, right? There are direct connections to projections and to mapping onto other basis vectors that come through when standard matrix layout can be used.Questions to: in last week's lab, there is no calculus in this notebook. I wish Mathematica always allowed straightforward matrix and vector multiplications, without behaving in a non-standard way if you have a column vector on the right, for example. Īm I wrong, or isn't this form of term-by-term multiplication when the second vector is actually a vector of basis vectors, a quick way to map onto a new basis? I just wish we could also use the intuitive notation or, even better. I wouldn't have guessed that you could pass in one 3D vector function, not a list of functions for each coordinate. Finance, Statistics & Business Analysis.Wolfram Knowledgebase Curated computable knowledge powering Wolfram|Alpha. Wolfram Universal Deployment System Instant deployment across cloud, desktop, mobile, and more. Wolfram Data Framework Semantic framework for real-world data.
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