Chapter 13: Q8TF (page 830)
The work done by a conservative force field in moving a particle around a closed path is zero.
The given statement is true.
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Evaluate the line integral, where C is the given curve.
\(\) \(\int_{\rm{C}} {\rm{x}} {{\rm{y}}^{\rm{4}}}{\rm{ds}}\), C is the right half of the circle \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ = 16}}\).
Use the Divergence Theorem to evaluate , where \({\bf{F}}(x,y,z) = {z^2}x{\bf{i}} + \left( {\frac{1}{3}{y^3} + \tan z} \right){\bf{j}} + \left( {{x^2}z + {y^2}} \right){\bf{k}}\)and \(S\) is the top half of the sphere \({x^2} + {y^2} + {z^2} = 1\) . (Hint: Note that \(S\) is not a closed surface. First compute integrals over \({S_1}\) and \({S_2}\), where \({S_1}\) is the disk , oriented downward, and \({S_2} = S \cup {S_1}\).)
\(\int\limits_{\rm{C}} {} {\rm{(y + z)dx + (x + z)dy + (x + y)dz}}\)
\({\rm{C}}\) consists of line segments from \({\rm{(0,0,0)}}\) to \({\rm{(1,0,1)}}\) and from \({\rm{(1,0,1)}}\) to \({\rm{(0,1,2)}}\).
Find the value of \(\iint_{S}{f}(x,y,z)dS \).
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