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Chapter 11: Problem 13

Find a parametric representation of the surface. $$z=3 x+4 y$$

Short Answer

Expert verified
The parametric representation of the surface \(z=3x+4y\) is \(x=u\), \(y=v\), \(z=3u + 4v\) where \(u\) and \(v\) are parameters.

Step by step solution

01

Identify The Form

First, we need to observe the general form of the equation given. This is a surface in \(\mathbb{R}^3\) represented as \(z=f(x,y)\). We need to express this in parametric form, i.e., \(x=x(u,v)\), \(y=y(u,v)\), \(z=z(u,v)\).
02

Assign The Parameters

Here we assign \(x = u\) and \(y = v\) where \(u\) and \(v\) are parameters. This is because in this case, it is most convenient to parameterize \(x\) and \(y\) directly as \(u\) and \(v\).
03

Express in Terms of Parameters

Substituting \(u\) for \(x\) and \(v\) for \(y\) in the equation \(z=3x+4y\), we get \(z=3u+4v\).
04

Formulate The Parametric Representation

Now, we have the surface represented parametrically as \(x = u\), \(y = v\), \(z = 3u + 4v\).

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Most popular questions from this chapter

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