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Let \(C\subset \mathbb {R}^3\) be the intersection of the cylinder \(x^2 + y^2 = 16\) and plane \(z=2\), oriented counterclockwise when
viewed from above. Compute
Let \(C\subset \mathbb {R}^3\) be the trajectory that travels along straight lines through the points \((0,1,0)\), \((4,1,0)\), \((4,1,2)\), \((0,1,2)\), and
back to \((0,1,0)\). Compute
\[ \int _C \langle y^2 + z, x y, x - z\rangle \cdot d \gamma \]
Let \(C\subset \mathbb {R}^3\) be the intersection of the cylinder \(y^2 + z^2 = 16\) and plane \(x=5\), oriented clockwise when seen
from the tip of the \(x\)-axis. Compute
Let \(C\subset \mathbb {R}^3\) be the curve that travels along straight lines first from \((3,0,0)\) to \((0,5,0)\), then from \((0,5,0)\) to \((0,0,15)\),
and then from \((0,0,15)\) to \((3,0,0)\). Compute
\[ \int _C \langle 2y, -x + z, y \rangle \cdot d \gamma \]
Let \(C\subset \mathbb {R}^3\) be the curve that goes from \((2,0)\) to \((-2,0)\) along the arc \(x^2 + y^2 = 4\), \(y \geq 0\), in the \(xy\)-plane, followed by
the curve that goes back to \((2,0)\) along the parabola \(z = x^2 - 4 \) in the \(xz\)-plane. Compute
\[ \int _C \langle x + y , z + x , y + z \rangle \cdot d \gamma \]
Let \(C_1\subset \mathbb {R}^3\) be the circle \(x^2 + y^2 = 1\) in the \(xy\)-plane, oriented counterclockwise, and \(C_2\subset \mathbb {R}^3\) the circle \(x^2 + y^2 = 1\) in the
plane \(z = 3\), oriented counterclockwise. Let \(F: \mathbb {R}^3 \to \mathbb {R}^3\) be a vector field with
\[ \int _{C_1} F \cdot d \gamma _1 \,\,\,\,\,\,\,\,\,\,\, \text { or } \,\,\,\,\,\,\,\,\,\,\, \int _{C_2} F \cdot d \gamma _2 \]
Let \(C_1\subset \mathbb {R}^3\) be the circle \(x^2 + y^2 = 1\) in the \(xy\)-plane, oriented counterclockwise, and \(C_2\subset \mathbb {R}^3\) the circle \(x^2 + y^2 = 16\) in the
\(xy\)-plane, oriented counterclockwise. Let \(F: \mathbb {R}^3 \to \mathbb {R}^3\) be a vector field with
\[ \int _{C_1} F \cdot d \gamma _1 \,\,\,\,\,\,\,\,\,\,\, \text { or } \,\,\,\,\,\,\,\,\,\,\, \int _{C_2} F \cdot d \gamma _2 \]
Let \(C_1\subset \mathbb {R}^3\) be the curve that travels along straight lines from \((4,0,0)\), to \((0,4,0)\), from \((0,4,0)\) to \((0,0,4)\),
and from \((0,0,4) \) to \((4,0,0)\). Let \(C_2\subset \mathbb {R}^3\) be the curve that travels along straight lines from \((1,0,0)\), to \((0,1,0)\),
from \((0,1,0)\) to \((0,0,1)\), and from \((0,0,1) \) to \((1,0,0)\). Let \(F: \mathbb {R}^3 \to \mathbb {R}^3\) be a vector field with
\[ \text {Curl}(F) (x,y,z) = \langle x, y, z \rangle . \]
Which one is larger?
\[ \int _{C_1} F \cdot d \gamma _1 \,\,\,\,\,\,\,\,\,\,\, \text { or } \,\,\,\,\,\,\,\,\,\,\, \int _{C_2} F \cdot d \gamma _2 \]
MTH 255H