Conservative Vector Fields:
1. Determine whether or not \({\bf F} \) is a conservative vector
field. (a) \( {\bf F}(x, y) = (\sin y +e^x
y){\bf i}+ (x \cos y + e^x ) {\bf j} \qquad \) (b) \( {\bf
F}(x, y) = \langle xy+y^2, x^2+2xy \rangle\)
solution
2. Show that the vector field \( {\bf F} = \langle \sin y + y e^x,
e^x+ x \cos y \rangle \) is conservative. Then find a potential function
\( f \), that is, find \( f(x,y) \) such that \(\nabla f = {\bf F} \).
solution
3. Show that the vector field \( {\bf F} = \langle \sin y - y \sin x,
\cos x+ x \cos y - y \rangle \) is conservative. Then find a potential function
\( f \), that is, find \( f(x,y) \) such that \(\nabla f = {\bf F} \).
solution
4. Show that the vector field \( {\bf F}(x, y) = \langle 2xy^3,
3x^2y^2 \rangle\) is conservative. Then evaluate the line integral
\(\displaystyle{ \int_C {\bf F} \cdot d{\bf r}} \) where \(C\) is any
smooth path joining the points \( (0, 0)\) and \((1, 1)\).
solution
5. Evaluate \(\displaystyle{ \int_C {\bf F} \cdot d{\bf r}} \) where
\( {\bf F} = \langle 1+2xy^2, 2x^2y \rangle \), and \(C\) is the
arc of the hyperbola \( y=1/x\) from \( (1, 1)\) to \((5,
1/5)\). solution
6. Compute \( \displaystyle{\int_C \vec{F}\cdot \vec{dr} }\) where
\(\vec{F}(x,y)=\langle x^3-xy,-\frac{1}{2}x^2+2y\rangle\) and \(C\) is
any path from (1, 2) to (3, 5).
solution
7. Find the work done by the force field \({\bf F}(x,y) =
\langle3\sqrt{x}y, 2x^{3/2} \rangle\) in moving an object on a plane
from (1, 2) to (4, 4).
solution
8. Find the work done \(\displaystyle{ \int_C {\bf F} \cdot
d{\bf r}} \) by \( {\bf F} = \langle y^2+2xe^y+1, 2xy+x^2e^y+2y
\rangle \), in moving an object along the path \( {\bf r}(t)=\sin
t \, {\bf i} + \cos t \, {\bf j} , \: 0 \leq t \leq \pi/2.\)
solution