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Chapter 14: Q. 58 (page 1121)

Show that the two definitions of dS in Definition 14.6 are equivalent, by showing that if S is a surface described by $r (x, y) = (x, y, z (x, y)) for (x, y) 鈭� D$ , then
$鈭�r_{x} (x, y) 脳 r_{y} (x, y)鈭� = \sqrt{{(\frac{鈭� z}{鈭� x})}^{2} + {(\frac{鈭� z}{鈭� y})}^{2} + 1}$

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Step 1. Given information.

given, $鈭�r_{x} (x, y) 脳 r_{y} (x, y)鈭� = \sqrt{{(\frac{鈭� z}{鈭� x})}^{2} + {(\frac{鈭� z}{鈭� y})}^{2} + 1}$

Step 2.

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Most popular questions from this chapter

If S is parametrized by r(u, v), why is $d S = r_{u} 脳 r_{v}$ the correct factor to use to account for distortion of area?

$F (x, y, z) = c o s (x y z) i + j - y z k$ , where S is the portion of the surface with equation $z = y^{3} - y^{2}$ that lies above and/or below the rectangle determined by $鈭� 3 鈮� x 鈮� 2$ and $鈭� 1 鈮� y 鈮� 1$ in the xy-plane, with n pointing in the positive z direction.

Find ${鈭�}_{S} 鈥� 1 d S$ , where S is the portion of the surface determined by $z = x^{2} 鈭� \sqrt{3} y$ that lies above the region in the xy-plane bounded by the x-axis and the lines with equations $y = 2 x, x = 3$ .

$F (x, y, z) = 鉄� y z, x z, x y 鉄�$ , where S is the portion of the saddle determined by $z = x^{2} 鈭� y^{2}$ that lies above the region in thexy-plane bounded by the x-axis and the parabola with equation $y = 1 鈭� x^{2}$ .

Given a smooth surface S described as a function z = f(x, y), calculate the upwards-pointing normal vector for S.

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Show that the two definitions of dS in Definition 14.6 are equivalent, by showing that if S is a surface described by r(x,y)=(x,y,z(x,y))for(x,y)鈭�D, then rx(x,y)脳ry(x,y)=鈭�z鈭�x2+鈭�z鈭�y2+1

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Step 2.

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