Chapter 13: Q12TF (page 830)
Determine whether the statement is true or false. If it is true, explainWhy. If it is false, explain why or give an example that disproves theStatement.
There is a vector field \({\rm{F}}\) such that curl
\({\rm{F = xi + yj + zk}}\)
The vector field \(G\)does not exist.
FALSE
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To determine the value of for \({\rm{F}}(x,y,z) = xzi + xj + y{\rm{k}}\) is \(0\).
\({\bf{F}}(x,y,z) = {e^y}\tan z{\bf{i}} + y\sqrt {3 - {x^2}} {\bf{j}} + x\sin y{\bf{k}}\) , \(S\) is the surface of the solid that lies above the \(xy\)-plane and below the surface
\(z = 2 - {x^4} - {y^4}, - 1,,x,,1\)\(- 1,,y,,1\)
Use a graph of the vector field \({\bf{F}}\) and the curve \(C\) to guess whether the line integral of \({\bf{F}}\) over \(C\) is positive, negative, or zero. Then evaluate the line integral.\({\bf{F}}(x,y) = (x - y){\bf{i}} + xy{\bf{j}}\), \(C\) is the arc of the circle \({x^2} + {y^2} = 4\) traversed counterclockwise from \((2,0)\) to \((0, - 2)\).
(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 \(C\).
\({\bf{F}}(x,y,z) = yz{e^{xz}}{\bf{i}} + {e^{xz}}{\bf{j}} + xy{e^{xz}}{\bf{k}}\),
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