24 1. Introduction 1.6.7. Surface area. Consider a surface written in the vector form r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k , (1.62) where u and v are parameters. (a) Justify or motivate the surface-area formula S = ||ru × rv|| du dv . (1.63) (b) Show that the above surface-area formula can also be written as S = EG F 2 du dv , (1.64) where E, F , and G are the coefficients of the first fundamental form. (c) Write the surface z = f(x, y) (1.65) in vector form and show that the above formulas for area imply that S = Ω 1 + f 2 x + f 2 y dx dy . (1.66) 1.6.8. Surface area of a hyperbolic paraboloid. Consider the hyperbolic paraboloid r(u, v) = u i + v j + uv k . (1.67) Determine the surface area for that portion of the paraboloid that is specified by values of u and v that lie in the first quadrant of (u, v) parameter space between the positive u- and v-axes and the circle u2 + v2 = 1 . (1.68) 1.6.9. Surface area of a helicoid. Find the area of the portion of the helicoid r(u, v) = u cos v i + u sin v j + bv k (1.69) that is specified by 0 u a and 0 v 2π.
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