Divergence Theorem

1 Divergence Theorem

Let \(\Sigma \subset \mathbb {R}^3\) be the sphere \(x^2 + y^2 + z^2 = 9\), negatively oriented. Compute
\[ \iint _{\Sigma } \langle x^2 + 3z, 2y + xyz, 5z - x \rangle \cdot dS \]
Let \(\Sigma \subset \mathbb {R}^3\) be the boundary of the box \([-1,2]\times [0,3]\times [-2,1]\), positively oriented. Compute
\[ \iint _{\Sigma } \langle 2x - yz, 3y + xz, 2yz + e^{x} \rangle \cdot dS \]
Let \(\Sigma \subset \mathbb {R}^3\) be the boundary of the solid bounded by the cylinder \(x^2 + y^2 = 1\) and the planes \(z=0\) and \(z=1\), oriented outward. Compute
\[ \iint _{\Sigma } \langle xz, yz, x^2 + y^2 \rangle \cdot dS \]
Let \(\Sigma \subset \mathbb {R}^3\) be the ellipsoid \(\dfrac {x^2}{4} + \dfrac {y^2}{9} + z^2 = 1\), oriented outward. Compute
\[ \iint _{\Sigma } \langle 2x + yz, 3y + x \cos z, e ^x + y^2 \rangle \cdot dS \]
Let \(\Sigma \subset \mathbb {R}^3\) be the closed surface consisting of the portion of the paraboloid \(z = 9 - x^2 - y^2\) with \(z \geq 0\), together with the disk \(x^2 + y^2 \leq 9\) in the \(xy\)-plane, oriented outward. Compute
\[ \iint _{\Sigma } \langle xy + 1, yz - 3, zx + 5 y \rangle \cdot dS \]
Let \(\Sigma \subset \mathbb {R}^3\) be the closed surface consisting of the portion of the cone \(z = \sqrt {x^2 + y^2}\) below the plane \(z = 4 \) together with the disk \(x^2 + y^2 \leq 16\) in the plane \(z=4\), oriented outward. Compute
\[ \iint _{\Sigma } \langle xy + 3 y, 4 y + z , 5z + x \rangle \cdot dS \]
Let \(\Sigma \subset \mathbb {R}^3\) be the boundary of the tetrahedron with vertices \((3,0,0)\), \((0, -3, 0)\), \((0,0,2)\), \((0,0,0)\), oriented outward. Compute
\[ \iint _{\Sigma } \langle 3x - 2y + e^z , 5x^2 - y , z - \cos y \rangle \cdot dS \]
Let \(\Sigma _1 \subset \mathbb {R}^3\) be the disk \(x^2 + y^2 \leq 1\) in the \(xy\)-plane, oriented upward, and \(\Sigma _2 \subset \mathbb {R}^3\) the portion of the sphere \(x^2 + y^2 + z^2 = 1 \) with \(z \geq 0\), oriented upward. Consider a vector field \(F : \mathbb {R}^3 \to \mathbb {R}^3\) with positive divergence. Which one is larger?
\[ \iint _{\Sigma _1} F \cdot dS \,\,\,\,\,\,\,\,\, \text { or } \,\,\,\,\,\,\,\,\, \iint _{\Sigma _2} F \cdot dS \]
Let \(\Sigma _1 \subset \mathbb {R}^3\) be the disk \(y^2 + z^2 \leq 1\) in the \(yz\)-plane, oriented towards the \(x\)-axis, and \(\Sigma _2 \subset \mathbb {R}^3\) the portion of the sphere \(x^2 + y^2 + z^2 = 1 \) with \(x \leq 0\), oriented towards the \(x\)-axis. Consider a vector field \(F : \mathbb {R}^3 \to \mathbb {R}^3\) with negative divergence. Which one is larger?
\[ \iint _{\Sigma _1} F \cdot dS \,\,\,\,\,\,\,\,\, \text { or } \,\,\,\,\,\,\,\,\, \iint _{\Sigma _2} F \cdot dS \]
Give a proof of the Divergence Theorem in the case where the region is rectangular. That is, let \(E = [a_1, b_1] \times [a_2, b_2] \times [a_3, b_3] \subset \mathbb {R}^3\), \(\partial E\) its boundary oriented positively, and \(F (x,y,z)\) a vector field whose domain contains \(E\). Then
\[ \iint _{\partial E } F \cdot dS = \iiint _E \text {div} (F) \, dV \]