Forms HW

1 Forms HW

Consider the following \(1\)-form in \(\mathbb {R}^2\):
\[ \omega = x^2 e ^y \, dy . \]
Compute and simplify \(d\omega \).
Consider the following \(1\)-form in \(\mathbb {R}^2\):
\[ \omega = 3xy \, dx + \cos x \, dy . \]
Compute and simplify \(d\omega \).
Consider the following \(1\)-form in \(\mathbb {R}^3\):
\[ \omega = xe^y \, dx + 5 y^2 \sin z \, dy + 7xz \, dz . \]
Compute and simplify \(d\omega \).
Consider the following \(1\)-form in \(\mathbb {R}^3\):
\[ \omega = x^2 \, dy - 3 y \, dz . \]
Compute and simplify \(d\omega \).
Consider the following \(2\)-form in \(\mathbb {R}^3\):
\[ \omega = xyz^2 \, dy \wedge dz + 2 \sin x \cos y \, dz \wedge dx + y e^z \, dx \wedge dy . \]
Compute and simplify \(d\omega \).
Consider the following \(1\)-form in \(\mathbb {R}^2\):
\[ \omega = 2x \, dx + \cos y \, dy . \]
Let \(C\subset \mathbb {R}^2 \) be the straight line from \((-1,0)\) to \((3 , \pi / 2)\). Compute
\[ \int _C \omega \, d \gamma . \]
Consider the following \(1\)-form in \(\mathbb {R}^3\):
\[ \omega = e^x \, dx + \cos z \, dz . \]
Let \(C\subset \mathbb {R}^3 \) be the portion of the circle \(z = \sqrt { 1 - x^2 }\) in the plane \(y = 0\) from \((1,0,0)\) to \((0,0,1)\). Compute
\[ \int _C \omega \, d \gamma . \]
Consider the following \(2\)-form in \(\mathbb {R}^3\):
\[ \omega = e^x \, dy \wedge dz + (5 - y ) \, dz \wedge dx + x^3 \cos y \, dx \wedge dy . \]
Let \(\Sigma \subset \mathbb {R}^3 \) be the square with vertices \((0,0,0)\), \((1,0,0)\), \((1,0,1)\), \((0,0,1)\), oriented towards the \(y\)-axis. Compute
\[ \iint _{\Sigma } \omega \, d S . \]