Chapter 13: Q5TF (page 830)
If \(F=Pi+Qj\)and \({{P}_{y}}={{Q}_{x}}\)in an open region D, then F is conservative.
The given statement is false.
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Evaluate the line integral\(\int_{\rm{C}} {\rm{F}} {\rm{ \times dr}}\) where \({\rm{C}}\)is given by the vector function\({\rm{r(t)}}\).
\({\rm{F(x,y) = xyi + 3}}{{\rm{y}}^{\rm{2}}}{\rm{j}}\)
Let \({\rm{F}}\)be the vector field shown in the figure.
(a) If \({{\rm{C}}_{\rm{1}}}\)is the vertical line segment from\({\rm{( - 3, - 3)}}\)to\({\rm{( - 3,3)}}\), determine whether \(\int_{{{\rm{C}}_{\rm{1}}}} {\rm{F}} \cdot {\rm{dr}}\) is positive, negative, or zero.
(b) If \({{\rm{C}}_{\scriptstyle{\rm{2}}\atop\scriptstyle}}\)is the counterclockwise-oriented circle with a radius of 3 and center the origin, determine whether \(\int_{{{\rm{C}}_{\rm{2}}}} {\rm{F}} \cdot {\rm{dr}}\)is positive, negative, or zero.
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 \(0 \le y \le 1.\)
Find the value of\(\iint_S {{x^2}}yzdS\)
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