Engineering Calculus Notes 420

Engineering Calculus Notes 420 - = k in the circle centered...

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408 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION x y z Figure 3.43: Vertical Sections x y z Figure 3.44: The cone x 2 + y 2 = z 2 (b) When there is a nonzero constant term, we will take it to be ± 1, and this leads to the possible equations x 2 ± y 2 = z 2 ± 1 . Again, up to interchange of variables, there are two possible shapes, which can be modelled by the equation above with the coe±cient of y 2 positive. The surface given by x 2 + y 2 = z 2 + 1 (3.48) intersects the horizontal plane z
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Unformatted text preview: = k in the circle centered on the z-axis of radius √ 1 + k 2 (Figure 3.45 ) To see how they ²t together, we consider the intersection of the surface with the two vertical coordinate planes, which are both hyperbolas opening horizontally (Figure 3.46 ). The resulting surface (Figure 3.47 ) is called a hyperboloid of one sheet (Figure 3.47 ). (c) The surface given by x 2 + y 2 = z 2 − 1 (3.49)...
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