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Engineering Calculus Notes 286

Engineering Calculus Notes 286 - yz-plane we get a similar...

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274 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Our second example is the surface given by the equation x 2 + y 2 z 2 = 1 . The intersection with any horizontal plane z = c is a circle x 2 + y 2 = c 2 + 1 of radius r = c 2 + 1 about the origin (actually, about the intersection of the plane z = c with the z -axis). Note that always r 1; the smallest circle is the intersection with the xy -plane. If we slice along the xz -plane y = 0 we get the hyperbola x 2 z 2 = 1 whose vertices lie on the small circle in the xy -plane. Slicing along the
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Unformatted text preview: yz-plane we get a similar picture, since x and y play exactly the same role in the equation. The shape we get, like a cylinder that has been squeezed in the middle, is called a hyperboloid of one sheet (Figure 3.9 ). Now, let us simply change the sign of the constant in the previous equation: x 2 + y 2 − z 2 = − 1 . The intersection with the horizontal plane z = c is a circle x 2 + y 2 = c 2 − 1...
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