Chapter 13: Q19E (page 771)
Evaluate the line integral\(\int_{\rm{C}} {\rm{F}} {\rm{ \times dr}}\) where \({\rm{C}}\)is given by the vector function\({\rm{r(t)}}\).
\({\rm{F(x,y) = xyi + 3}}{{\rm{y}}^{\rm{2}}}{\rm{j}}\)
The value of the line integral is \({\rm{45}}\).
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(a) Find a function \(f\) such that \({\bf{F}} = \nabla f\) and (b) use part (a) to evaluate \(\int_C {\bf{F}} \cdot d{\bf{r}}\) along the given curve \(C\).
\({\bf{F}}(x,y,z) = yz{\bf{i}} + xz{\bf{j}} + (xy + 2z){\bf{k}}\), \(C\) is the line segment from \((1,0, - 2)\) to \((4,6,3)\)
Determine whether of not \({\bf{F}}\) is a conservative vector field. If it is, find a function \(f\) such that\({\bf{F}} - \nabla f\).
\({\bf{F}}(x,y) - \left( {2xy + {y^{ - 2}}} \right){\bf{i}} + \left( {{x^2} - 2x{y^{ - 3}}} \right)\)
Find the value of \(\iint_S ydS\)\(z = \frac{2}{3}\left( {{x^{\frac{3}{2}}} + {y^{\frac{3}{2}}}} \right),0 \le x \le 1\)and \(0 \le y \le 1.\)
(a) Find a function \(f\) such that \({\bf{F}} = \nabla f\) and (b) use part (a) to evaluate \(\int_C {\bf{F}} \cdot d{\bf{r}}\) along the given curve \(C\).
\({\bf{F}}(x,y) = (1 + xy){e^{xy}}{\bf{i}} + {x^2}{e^{xy}}{\bf{j}}\),
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