Green’s Theorem, Divergence Theorem, and Stokes’ Theorem

Green’s Theorem

We will start with the following 2-dimensional version of fundamental theorem of calculus:

Green’s Theorem Let \(D\) be the region in the plane bounded by piecewise smooth simple closed curve \(C\). Assume that \(C\) is oriented counter-clockwise. Let \(\overrightarrow F=\langle P,Q\rangle\) be the vector field defined on \(D\) such that \(P\) and \(Q\) are differentiable real-valued functions. The following equality holds: \[\oint_C\overrightarrow F\cdot \,d\overrightarrow r=\iint_D \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dxdy.\]

Divergence theorem

The following theorem provides a relation between triple integrals and surface integrals over the closed surfaces.

Divergence Theorem (Theorem of Gauss and Ostrogradsky) Let \(S\) be a closed orientable surface that encloses the solid \(G\). Assume that \(S\) is oriented outwards. Let \(\overrightarrow F:G\to\mathbb R^3\) be a vector fields whose components are continuously differentiable functions on \(G\). Then the following equality holds: \[\iint_S \overrightarrow F\cdot d\overrightarrow S=\iiint_G \nabla\cdot \overrightarrow F\,dV.\]

Example Assume that \(T\) is a solid with boundary \(S\). Assume that \(u\) and \(v\) are two times continuously differentiable functions such that \(v=0\) on the set \(S\) and \(u\) satisfies \[u_{xx}+u_{yy}+u_{zz}=f(x,y,z).\] Prove that \[\iiint_T \nabla u\cdot\nabla v\,dxdydz= -\iiint_T fv\,dxdydz.\]

Stokes’ Theorem

Stokes’ Theorem Assume that \(\sigma\) is an oriented piecewise smooth surface in space whose boundary is a piecewise smooth curve \(\gamma\). Assume that the curve \(\gamma\) is oriented positively with respect to \(\sigma\). Let \(\overrightarrow F:\sigma\to\mathbb R^3\) be a vector field with continuously differentiable components on \(\sigma\). Then the following equality holds: \[\iint_{\sigma}\left(\nabla\times\overrightarrow F\right)\cdot d\overrightarrow S=\int_{\gamma} \overrightarrow F\cdot d\overrightarrow r.\]

One of the conditions of the theorem requires the curve to be oriented positively with respect to the surface. The meaning is the following: If a person stands on the surface parallel to the normal vector and walks along the boundary in the direction of the curve, the surface should be under the person’s left arm.

Example Let \(\overrightarrow F=\langle z,x,2x\rangle\) and let \(G\) be the part of the paraboloid \(z=9-x^2-y^2\) located above the plane \(z=5\). Verify the Stokes’ theorem by evaluating:
  • (a) \(\displaystyle \iint_G \left(\nabla \times \overrightarrow F\right)\cdot d\overrightarrow S\), where the orientation is such that the unit normal vector to \(G\) points upward at the point \((0,0,9)\).
  • (b) \(\displaystyle \int_{\gamma} \overrightarrow F\cdot d\overrightarrow r\), where \(\gamma\) is the boundary of \(G\) oriented counter-clockwise for the person standing at \((0,0,100)\).

Practice problems

Problem 1. Evaluate the integral \[\int_C \overrightarrow F\cdot d\overrightarrow r,\] where \(C\) is the curve with the parametrization \(\overrightarrow r(t)=\langle \cos t, \sin t, \sin\frac{t}2\rangle\) for \(0\leq t\leq 2\pi\) and \(\overrightarrow F\) is the vector field defined by \(\overrightarrow F=\left\langle \sqrt{x^4+2}+2y, \sqrt{y^4+3}-2x, \sqrt{z^4+4}+2z\right\rangle\).

Problem 2. Let \(T\) be the solid cylinder \(x^2 + y^2 \leq 9\), \(0 \leq z \leq 2\), and let \(S\) be the surface forming the boundary of \(T\). Consider the vector field \(\overrightarrow F = \langle x, y,z\rangle\). Assume that \(S\) is oriented outwards. Evaluate \(\iint_S\overrightarrow F\cdot d\overrightarrow S\).

Problem 3. Evaluate the integral \[\int_C y^2\,dx+2xy\,dy+z\,dz\] where \(C\) is the intersection of the paraboloids \(z=x^2+4y^2\) and \(y=25-x^2-4z^2\) oriented counter-clockwise when viewed from the origin.

Problem 4. Let \(E\) be the surface of the ellipsoid \(\displaystyle \frac{x^2}{9}+\frac{y^2}{25}+\frac{z^2}4=1\). Let \(\vec F=\langle y,x,z^3\rangle\). Evaluate the integral \(\displaystyle \iint_E \vec F\cdot\vec n\,dS\), where \(\vec n\) is the outward unit normal vector.

Problem 5. Let \(\vec F=\langle xy,4x+6y\rangle\) and let \(\gamma\) be a counter-clockwise oriented curve that consists of the segment on \(x\)-axis from \(-3\) to \(3\) and of the graph of the function \(y=\sqrt{9-x^2}\) for \(x\in[-3,3]\). Use Green’s theorem to evaluate the integral \(\displaystyle \oint_{\gamma}\vec F\cdot d\vec r\).