Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2022
Sporkz
Views
59
Visualizing surfaces in three dimensions. Review: Trying to draw a function of two variables As a first step, we can plot the graph of f, which is a function of two variables. To determine the possible values of x and y, we can construct a graph of this equation. Aswith equations involving a single variable, the possible values of x and y depend on theshape of this graph.We plot a point for each of them in the xy- plane, whose height is the value of a function ofthese parameters. So we'll plot— let's say z equals f(x, y)—and now that will actuallybecome a surface in space. So for each value of x and y, here we have (x, y) in the xy-plane, then we'll plot the point in space at position (x y, z) equals f(x, y).If we combine all these points together, they will give us a surface that sits in space. Given a function of two variables, how do we represent it geometrically? Let's take our firstexample. We are given the function f of (x, y) equal to negative y. However, it does notdepend on x, which is not a problem. It is still a valid function of x and y.
The function z=−y is constant with respect to x. Because the curve is a plane, we can drawits graph by examining the plane in space defined by z being equal to −y. If we want to drawthe graph of z = −y, we can use the axes shown on the blackboard. If we look at whathappens in the yz-plane, which is the plane of the blackboard, we will see that it will look likea line that goes downward with slope 1. If we change x, nothing happens because x does not appear in this equation. If instead ofsetting x equal to 0, we set x equal to 1 or minus 1, it still looks exactly the same. Now wehave a plane that contains the x-axis and slopes downward with a slope of 1. It's hard todraw, but you can see immediately what the big problem with graphs will be: They are hardto read.
Visualizing 3D Surfaces: Exploring Functions of Two Variables
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