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Prove that 1+ ∂g ∂x 2 + ∂g ∂y 2 = y x 2 5 + , 4 and then compute the surface integral F x dS. Brain power Please click the advert By 2020, wind could provide one-tenth of our planet’s electricity needs. Already today, SKF’s innovative knowhow is crucial to running a large proportion of the world’s wind turbines. Up to 25 % of the generating costs relate to maintenance. These can be reduced dramatically thanks to our systems for on-line condition monitoring and automatic lubrication. We help make it more economical to create cleaner, cheaper energy out of thin air.
I 1) We get by a squaring and a rearrangement that M is a subset of the point set given by (z + a)2 − 2 = a2 . This describes in the whole P Z-plane an hyperbola of centrum (0, −a) and half axes a and a. 2) The line element on M is given by ds = 1+ = 1+ d dz 2 dz = 1+ z 2 + 2az + a2 dz = z 2 + 2az 2z + 2a √ 2 z 2 + 2az 2 dz 2z 2 + 4az + a2 dz. com 49 Calculus 2c-8 Surface integral 3) First express the integrand in semi polar coordinates on the surface: f (x, y, z) = |x|(z + a) x2 + y2 = | cos ϕ| (z + a) = | cos ϕ|(z + a).
I The surface element is dS = P dϕ ds, where P = (z) = ds = 1+ d dz 2 dz = 1+ 1 1 ·√ 2 z √ z and 2 dz, thus F 1 √ dS 1 + 4z 4 = 2π 0 4 = 2π 0 √ 1 1 · z· 1+ dz 1 + 4z 4z √ 2π z 1 + 4z · dz = 1 + 4z 4z 2 4 dz = 4π. 8 A surface of revolution F is given in semi polar coordinates ( , ϕ, z) by z= 3 ∈ 0, , 1 , 2 ϕ ∈ [0, 2π]. Sketch the meridian curve M, and ﬁnd the line element ds on this curve. Then compute the surface integral 2 F 1 + 9z dS. A Surface integral. D Follow the guidelines. 1 Figure 25: The meridian curve M.