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Section 5-6 : Conservative Vector Fields

For problems 1 – 4 determine if the vector field is conservative.

  1. \(\vec F = \left( {2x{y^3} + {{\bf{e}}^x}\cos \left( y \right)} \right)\vec i + \left( {{{\bf{e}}^x}\sin \left( y \right) - 3{x^2}{y^2}} \right)\vec j\)
  2. \(\vec F = \left( {x{y^2} - 3{y^4} + 2} \right)\vec i + \left( {x{y^2} + {x^2}{y^2} - x} \right)\vec j\)
  3. \(\displaystyle \vec F = \left( {2 + 12x{y^2} - 3{x^2}\sqrt y } \right)\vec i - \left( {\frac{{{x^3}}}{{2\sqrt y }} - 12{x^2}y} \right)\vec j\)
  4. \(\displaystyle \vec F = \left( {8 - \frac{{3{x^2}}}{y} + 5{x^4}{y^2}} \right)\vec i + \left( {6 + \frac{{{x^3}}}{{{y^2}}} - 3{y^2} + 2{x^5}y} \right)\vec j\)

For problems 5 – 11 find the potential function for the vector field.

  1. \( \displaystyle \vec F = \left( {4{x^3} + 3y + \frac{{2{y^3}}}{{{x^3}}}} \right)\vec i + \left( {3x - 3{y^2} - \frac{{3{y^2}}}{{{x^2}}}} \right)\vec j\)
  2. \(\vec F = \left( {3{x^2}{{\bf{e}}^{2y}} + 4y{{\bf{e}}^{4x}}} \right)\vec i - \left( {7 - 2{x^3}{{\bf{e}}^{2y}} - {{\bf{e}}^{4x}}} \right)\vec j\)
  3. \(\vec F = \left( {\cos \left( x \right)\cos \left( {x + y} \right) - 2{y^2} - \sin \left( x \right)\sin \left( {x + y} \right)} \right)\vec i - \left( {4xy + \sin \left( x \right)\sin \left( {x + y} \right)} \right)\vec j\)
  4. \(\displaystyle \vec F = \left( {\frac{4}{{{x^2}}} + \frac{{2x}}{y} + \frac{2}{{{x^2}{y^3}}}} \right)\vec i + \left( {\frac{6}{{x{y^4}}} - \frac{{1 + {x^2}}}{{{y^2}}}} \right)\vec j\)
  5. \(\vec F = \left( {2x{{\bf{e}}^{{x^{\,2}} - z}}\sin \left( {{y^2}} \right) - 3{y^3}} \right)\vec i + \left( {2y{{\bf{e}}^{{x^{\,2}} - z}}\cos \left( {{y^2}} \right) - 9x{y^2}} \right)\vec j + \left( {12z - {{\bf{e}}^{{x^{\,2}} - z}}\sin \left( {{y^2}} \right)} \right)\vec k\)
  6. \( \displaystyle \vec F = \left( {12x - 5{z^2}} \right)\vec i + \ln \left( {1 + {z^2}} \right)\vec j - \left( {10xz - \frac{{2yz}}{{1 + {z^2}}}} \right)\vec k\)
  7. \(\vec F = \left( {z{y^2}{{\bf{e}}^{y - x}} - x{y^2}z{{\bf{e}}^{y - x}}} \right)\vec i + \left( {2xyz{{\bf{e}}^{y - x}} + x{y^2}z{{\bf{e}}^{y - x}}} \right)\vec j + \left( {x{y^2}{{\bf{e}}^{y - x}} - 24z} \right)\vec k\)
  8. 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 \(\left( {4,3} \right)\).
  9. 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.
  10. 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 \(x = 1\).
  11. 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 \(\left( {2, - 4,3} \right)\).
  12. Evaluate \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F\left( {x,y} \right) = \left( {12xy - 2x} \right)\,\vec i + \left( {6{x^2} - 8xy} \right)\vec j + \left( {8 - 4{y^2}} \right)\vec k\) 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 \(0 \le t \le 6\).
  13. 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.
    This curve has three line segments in it.  The first line segment starts at (-1,2) and ends at the origin.  The second line segment starts at the origin and ends at (2,1).  The final line segment starts at (2,1) and ends at (4,-2).
  14. 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.
    This curve starts with the upper half of the circle of radius 1 centered at the origin with clockwise rotation (i.e. starts at (-1,0) and ends at (1,0)).  It is followed by the portion of $y=\sin(\pi x)$ starting at (1,0) and ending at (2,0).  The final portion of the curve is the lower half of the circle given by ${{\left( x-3 \right)}^{2}}+{{y}^{2}}=1$ with counter clockwise rotation (i.e. starts at (2,0) and ends at (4,0)).