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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 $r (x, y) = (x, y, z (x, y)) for (x, y) \in D$ , then
$∥r_{x} (x, y) \times r_{y} (x, y)∥ = \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1}$

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Step 1. Given information.

given, $∥r_{x} (x, y) \times r_{y} (x, y)∥ = \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1}$

Step 2.

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Most popular questions from this chapter

$C o m p u t e t h e c u r l o f t h e v e c t o r f i e l d s i n E x e r c i s e s 23 - 28 . G (x, y, z) = (x^{2} + y - z) i - 2 y \cos zj + e^{x^{2} + y^{2} + z^{2}} k$

Make a chart of all the new notation, definitions, and theorems in this section, including what each new item means in terms you already understand.

Consider the vector field $F (x, y, z) = (y z, x z, x y)$ . Find a vector field $G (x, y, z)$ with the property that, for all points in role="math" localid="1650383268941" $ℝ^{3}, G (x, y, z) = 2 F (x, y, z)$ .

$f (x, y, z) = <\frac{1}{x \ln (z + 2 y)}>$ , where Sis the surface given by $r (u, v) = 2 u i + (u + v) j + (2 u - 2 v) k$ for $3 \leq u \leq 7$ and $2 \leq v \leq 10$ .

The current through a certain passage of the San Juan Islands in Washington State is given by

$F = <F_{1} (x, y), F_{2} (x, y)> = <0, 1.152 - 0.8 x^{2}> .$

Consider a disk R of radius 1 mile and centered on this region. Denote the boundary of the disk by localid="1650455155947" $\partial R$ .

(a) Compute localid="1650455166518" $\iint_{R} (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) d A$ .
(b) Show that $\int_{\partial R} F \cdot n d s = \int_{0}^{2 π} (1.152 - 0.8 \cos^{2} θ) \sin θ d θ .$ Conclude that Green's Theorem is valid for the current in this area of the San Juan Islands.
(c) What do the integrals from Green's Theorem tell us about this region of the San Juan Islands?

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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)∈D, then rx(x,y)×ry(x,y)=∂z∂x2+∂z∂y2+1

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Step 1. Given information.

Step 2.

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