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Let \(C \subset \mathbb {R}^2\) be the curve that travels along straight lines from \((0,0)\) to \((4,0)\), then from \((4,0) \) to \((0,3)\), and
then from \((0,3)\) back to \((0,0)\). Compute
\[ \int _C \langle 3x-y^2 - 1 , x + 2y \rangle \cdot d \gamma \]
The curve is closed and travels counterclockwise, so the integral above equals the
integral of \(1 + 2 y \) (the curl of \(\langle 3x-y^2 - 1 , x + 2y \rangle \)) over the region surrounded by \(C\).
Let \(C \subset \mathbb {R}^2\) be the curve that travels along straight lines from \((-1,2)\) to \((1,2)\), then from \((1,2) \) to \((1,3)\), then
from \((1,3) \) to \((-1,3)\), and then from \((-1,3)\) back to \((-1,2)\). Compute
\[ \int _C \langle x^2 y + 3x - 5, 2xy + e^y + 3 \rangle \cdot d \gamma \]
The curve is closed and travels counterclockwise, so the integral above equals the
integral of \( 2y - x^2 \) (the curl of \(\langle x^2 y + 3x - 5, 2xy + e^y + 3 \rangle \)) over the region surrounded by \(C\).
Let \(C \subset \mathbb {R}^2\) be the circle \(x^2 + y^2 = 9\) oriented counterclockwise. Compute
\[ \int _C \langle xy + 5y^3 + 2 , x - y e ^y \rangle \cdot d \gamma \]
Let \(C \subset \mathbb {R}^2\) be the ellipse \(\frac {x^2}{4} + \frac {y^2}{9} = 1\) oriented counterclockwise. Compute
\[ \int _C \langle -3y,\; 2x + y \rangle \cdot d \gamma \]
Let \(C \subset \mathbb {R}^2\) be the curve that travels along straight lines from \((-2,4)\) to \((-2,0)\), then from \((-2,0)\) to \((2,0)\), then
from \((2,0)\) to \((2,4)\), and then back to \((-2,4)\) along the parabola \(y = x^2\). Compute
\[ \int _C \langle 3 y + x^2 - 2, 2x + y + 7 \rangle \cdot d \gamma \]
The curve is closed and travels counterclockwise, so the integral above equals the
integral of \( 2-3 =- 1 \) (the curl of \(\langle 3 y + x^2 - 2, 2x + y + 7 \rangle \)) over the region surrounded by \(C\).
Let \(C \subset \mathbb {R}^2\) be the curve
that travels along straight lines from \((0,0)\) to \((3,1)\), then from \((3,1) \) to \((4,4)\), then from \((4,4) \) to \((1,3)\), and then
from \((1,3)\) back to \((0,0)\). Compute
\[ \int _C \langle y - 3 x + 2, x^2 - 3 y + 5 \rangle \cdot d \gamma \]
Let \(C_1 \subset \mathbb {R}^2\) be the curve that travels along straight lines from \((0,0)\) to \((3,0)\), then from \((3,0) \) to \((4,4)\). Let \(C_2 \subset \mathbb {R}^2\) be
the curve that travels along straight lines from \((0,0)\) to \((0,5)\), then from \((0,5) \) to \((4,4)\). Let \(C_3 \subset \mathbb {R}^2\) be the
straight line from \((0,0)\) to \((4,4)\). Let \(F : \mathbb {R}^2 \to \mathbb {R}^2\) be a vector field with positive curl. Among the integrals
\[ \int _{C_1} F\cdot d \gamma _1 , \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \int _{C_2} F\cdot d \gamma _2 , \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\, \int _{C_3} F\cdot d \gamma _3 , \]
which one is the largest and which one the smallest?
The curve \(C_4\) that travels along \(C_1\) and then along \(C_3\) but in the reverse direction, is closed
and travels counterclockwise. Therefore
\[ \int _{C_1} F \cdot d_{\gamma _1} - \int _{C_3} F \cdot d _{\gamma _3} = \iint _{D} \text {curl}(F) \, dA \]
