**Unformatted text preview: **y = x 2 + z 2 and in the half-space { ( x, y, z ) | y ≤ 1 } . We assume that the surface is oriented in the positive direction of the y-axis. 17.8b. Suppose S is a surface with boundary C . Show that if v is a ﬁxed vector and F ( x, y, z ) = ± x, y, z ² , then 2 ± ± S v · d S = ± C ( v × F ) · d r . 17.8c. Suppose F is a vector ﬁeld. Consider two surfaces S and S ± which possess the same boundary C . Use sketches (note plural) to describe how S and S ± must be oriented so that ± ± S curl( F ) · d S = ± ± S ± curl( F ) · d S ± . 17.8d. Suppose F is a vector ﬁeld. Expanding upon the above problem: Suppose that S is a closed surface (that is, S is a boundary for a region of space). Show that ± ± S curl( F ) · d S = 0 ....

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- Winter '08
- Fish
- Multivariable Calculus, Vector Calculus, Manifold, Surface, Vector field