Chapter 13: Q3E (page 805)
Identify the surface with the given vector equation
\(r(s,t) = \left( {s,t,{t^2} - {s^2}} \right)\)
The give vector equation surface represents a hyperbolic paraboloid
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\(\int\limits_{\rm{C}} {} {\rm{(y + z)dx + (x + z)dy + (x + y)dz}}\)
\({\rm{C}}\) consists of line segments from \({\rm{(0,0,0)}}\) to \({\rm{(1,0,1)}}\) and from \({\rm{(1,0,1)}}\) to \({\rm{(0,1,2)}}\).
(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 \(C\).
\({\bf{F}}(x,y) = (1 + xy){e^{xy}}{\bf{i}} + {x^2}{e^{xy}}{\bf{j}}\),
\({\bf{F}}(x,y,z) = {e^y}\tan z{\bf{i}} + y\sqrt {3 - {x^2}} {\bf{j}} + x\sin y{\bf{k}}\) , \(S\) is the surface of the solid that lies above the \(xy\)-plane and below the surface
\(z = 2 - {x^4} - {y^4}, - 1,,x,,1\)\(- 1,,y,,1\)
Determine whether of not \({\bf{F}}\) is a conservative vector field. If it is, find a function \(f\) such that\({\bf{F}} - \nabla f\).
\({\bf{F}}(x,y) - (xy\cosh xy + \sinh xy){\bf{i}} + \left( {{x^2}\cosh xy} \right){\bf{j}}\)
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