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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 16.6 : Conservative Vector Fields
For problems 1 – 4 determine if the vector field is conservative.
- \(\vec F = \left( {2x{y^3} + {{\bf{e}}^x}\cos \left( y \right)} \right)\vec i + \left( {{{\bf{e}}^x}\sin \left( y \right) - 3{x^2}{y^2}} \right)\vec j\)
- \(\vec F = \left( {x{y^2} - 3{y^4} + 2} \right)\vec i + \left( {x{y^2} + {x^2}{y^2} - x} \right)\vec j\)
- \(\displaystyle \vec F = \left( {2 + 12x{y^2} - 3{x^2}\sqrt y } \right)\vec i - \left( {\frac{{{x^3}}}{{2\sqrt y }} - 12{x^2}y} \right)\vec j\)
- \(\displaystyle \vec F = \left( {8 - \frac{{3{x^2}}}{y} + 5{x^4}{y^2}} \right)\vec i + \left( {6 + \frac{{{x^3}}}{{{y^2}}} - 3{y^2} + 2{x^5}y} \right)\vec j\)
For problems 5 – 11 find the potential function for the vector field.
- \( \displaystyle \vec F = \left( {4{x^3} + 3y + \frac{{2{y^3}}}{{{x^3}}}} \right)\vec i + \left( {3x - 3{y^2} - \frac{{3{y^2}}}{{{x^2}}}} \right)\vec j\)
- \(\vec F = \left( {3{x^2}{{\bf{e}}^{2y}} + 4y{{\bf{e}}^{4x}}} \right)\vec i - \left( {7 - 2{x^3}{{\bf{e}}^{2y}} - {{\bf{e}}^{4x}}} \right)\vec j\)
- \(\vec F = \left( {\cos \left( x \right)\cos \left( {x + y} \right) - 2{y^2} - \sin \left( x \right)\sin \left( {x + y} \right)} \right)\vec i - \left( {4xy + \sin \left( x \right)\sin \left( {x + y} \right)} \right)\vec j\)
- \(\displaystyle \vec F = \left( {\frac{4}{{{x^2}}} + \frac{{2x}}{y} + \frac{2}{{{x^2}{y^3}}}} \right)\vec i + \left( {\frac{6}{{x{y^4}}} - \frac{{1 + {x^2}}}{{{y^2}}}} \right)\vec j\)
- \(\vec F = \left( {2x{{\bf{e}}^{{x^{\,2}} - z}}\sin \left( {{y^2}} \right) - 3{y^3}} \right)\vec i + \left( {2y{{\bf{e}}^{{x^{\,2}} - z}}\cos \left( {{y^2}} \right) - 9x{y^2}} \right)\vec j + \left( {12z - {{\bf{e}}^{{x^{\,2}} - z}}\sin \left( {{y^2}} \right)} \right)\vec k\)
- \( \displaystyle \vec F = \left( {12x - 5{z^2}} \right)\vec i + \ln \left( {1 + {z^2}} \right)\vec j - \left( {10xz - \frac{{2yz}}{{1 + {z^2}}}} \right)\vec k\)
- \(\vec F = \left( {z{y^2}{{\bf{e}}^{y - x}} - x{y^2}z{{\bf{e}}^{y - x}}} \right)\vec i + \left( {2xyz{{\bf{e}}^{y - x}} + x{y^2}z{{\bf{e}}^{y - x}}} \right)\vec j + \left( {x{y^2}{{\bf{e}}^{y - x}} - 24z} \right)\vec k\)
- Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\displaystyle \vec F\left( {x,y} \right) = \left( {\frac{{3{x^2}}}{{y - 1}} - 3{x^2}y} \right)\,\vec i + \left( {8y - {x^3} - \frac{{{x^3}}}{{{{\left( {y - 1} \right)}^2}}}} \right)\vec j\) and C is the line segment from \(\left( {1,2} \right)\) to \(\left( {4,3} \right)\).
- Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = \left( {{y^2} - 4y + 5} \right)\,\vec i + \left( {2xy - 4x - 9} \right)\vec j\) and C the upper half of \(\displaystyle \frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{16}} = 1\) with clockwise rotation.
- Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\displaystyle \vec F\left( {x,y} \right) = - \left( {3 - \left( {1 + 2y} \right){{\bf{e}}^{x - 1}}} \right)\,\vec i + \left( {3{y^2} + 2{{\bf{e}}^{x - 1}}} \right)\vec j\) and C is the portion of \(y = {x^3} + 1\) from \(x = - 2\) to \(x = 1\).
- Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\displaystyle \vec F\left( {x,y,z} \right) = \frac{x}{{\sqrt {{x^2} + {z^2}} }}\,\vec i + \left( {2yz - 6y} \right)\vec j + \left( {{y^2} + \frac{z}{{\sqrt {{x^2} + {z^2}} }}} \right)\vec k\) and C is the line segment from \(\left( {1,0, - 1} \right)\) to \(\left( {2, - 4,3} \right)\).
- Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y,z} \right) = \left( {12xy - 2x} \right)\,\vec i + \left( {6{x^2} - 8yz} \right)\vec j + \left( {8 - 4{y^2}} \right)\vec k\) and C is the spiral given by \(\vec r\left( t \right) = \left\langle {\sin \left( {\pi t} \right),\cos \left( {\pi t} \right),3t} \right\rangle \) for \(0 \le t \le 6\).
- Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = \left( {8 - 14x{y^2} + 2y{{\bf{e}}^{2x}}} \right)\,\vec i + \left( {{{\bf{e}}^{2x}} - 14{x^2}y} \right)\vec j\) and C is the curve shown below.
- Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = \left( {6x - 5{y^2} + 2x{y^3} - 10} \right)\,\vec i + \left( {3{x^2}{y^2} - 10xy} \right)\vec j\) and C is the curve shown below.
