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標題:
Calculus III
發問:
最佳解答:
(a) S is given by the equation z = f(x,y) = √(1-y2) for each fixed x, the cross section of S on the plane x=c is √(1-y2) = z, i.e., 1-y2 = z2 y2 + z2 = 1 The last equation is a circle of radius 1. However, since by definition, z = √(1-y2) ≧ 0, hence the cross section is indeed only the upper half circle. This is true for all x, therefore, S is a surface with identical cross sections, which is an upper half circle. This means that S is the upper half of a circular cylinder of radius 1. (b) z = √(1-y2) z - √(1-y2) = 0. Take g(x,y,z) = z - √(1-y2), the S is the level surface g(x,y,z)=0
其他解答:
Calculus III
發問:
此文章來自奇摩知識+如有不便請留言告知
the surface S is the graph of f(x,y)=square root of (1-y^2). a) explain why S is the upper half of a circular cylinder of radius 1, centered along the x-axis b)find a level surface g(x,y,z)=c representing S.最佳解答:
(a) S is given by the equation z = f(x,y) = √(1-y2) for each fixed x, the cross section of S on the plane x=c is √(1-y2) = z, i.e., 1-y2 = z2 y2 + z2 = 1 The last equation is a circle of radius 1. However, since by definition, z = √(1-y2) ≧ 0, hence the cross section is indeed only the upper half circle. This is true for all x, therefore, S is a surface with identical cross sections, which is an upper half circle. This means that S is the upper half of a circular cylinder of radius 1. (b) z = √(1-y2) z - √(1-y2) = 0. Take g(x,y,z) = z - √(1-y2), the S is the level surface g(x,y,z)=0
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