Line integrals of vector fields

1 Line integrals of vector fields

Let \(C\subset \mathbb {R}^3\) be the curve with parametrization \(\gamma : [0,1 ] \to \mathbb {R}^3\) given by
\[ \gamma (t) = ( t + t^2 , t ^4 , 2t + t^2 - 3 ). \]
Compute
\[ \int _C \langle - y , x + 2 , z + 1 \rangle \cdot ds \]
Let \(C\subset \mathbb {R}^3\) be the curve with parametrization \(\gamma : [0, \pi ] \to \mathbb {R}^3\) given by
\[ \gamma (t) = ( \sin t , e ^t , \cos t ). \]
Compute
\[ \int _C \langle x + y , y + z , z + x \rangle \cdot ds \]
Let \(C\subset \mathbb {R}^3\) be the curve with parametrization \(\gamma : [0, 1 ] \to \mathbb {R}^3\) given by
\[ \gamma (t) = ( t ^2 \cos (4 \pi t ) , t^2 + t^3 , t^3 + t^4 + t^5 ). \]
Compute
\[ \int _C \, \left [ ( yz + 2x) \, dx + ( xz + z) \, dy + ( xy + y + 3 z^2 ) \, dz \right ] \]
Let \(C\subset \mathbb {R}^3\) be the piece of the parabola \(x = y^2 \) from \((0,0)\) to \((4,2)\). Compute
\[ \int _C \, \left [ e^ {y^2} \, dx + x \, dy \right ] \]
Let \(C \subset \mathbb {R}^3\) be the unit circle in the \(xy\)-plane travelled counterclockwise when viewed from above. Compute
\[ \int _C \langle e^{x^2} , \sin (y^2) , \cos (z^3) \rangle \cdot ds \]
Let \(C\subset \mathbb {R}^3\) be portion from \((-2,0,2)\) to \((2,0,-2)\) of the curve of intersection of the plane \(x + y + z = 0\) and the cylinder \(x^2 + z^2 = 8\). Compute
\[ \int _C \langle x + z , \cos y , e^z + x \rangle \cdot ds \]
Let \(C\subset \mathbb {R}^3\) be the curve with parametrization \(\gamma : [0, 1 ] \to \mathbb {R}^3\) given by
\[ \gamma (t) = ( t^2 + 2t - 1 , (3 + t) \sqrt { t^2 + 1 } , e^t + 2 t^2 ). \]
Compute
\[ \int _C \langle e ^x (z-1) , z \cos y , e ^x + \sin y \rangle \cdot ds \]
Let \(C\subset \mathbb {R}^3\) be the curve with parametrization \(\gamma : [0, \pi / 2 ] \to \mathbb {R}^3\) given by
\[ \gamma (t) = ( 2 \sin t , \sin t , 3\cos t ). \]
Compute
\[ \int _C \, \left [ yz \, dx + xy \, dy + xz \, dz \right ] \]