Green’s, Stokes’, and the Divergence Theorems

Exercises

  1. Evaluate \(\int_C x^4\,\mathrm{d}x + xy\,\mathrm{d}y,\) where \(C\) is the triangular curve consisting of the line segments from \((0,0)\) to \((1,0)\) and from \((1,0)\) to \((0,1)\) and from \((0,1)\) to \((0,0).\)
  2. Evaluate \(\oint \Big(3y-\mathrm{e}^{\sin(x)}\Big)\,\mathrm{d}x + \Big(7x+\sqrt{y^4+1}\Big)\,\mathrm{d}y,\) where \(C\) is the circle \(x^2+y^2=9.\)
  3. Calculate the area enclosed by the ellipse \(\frac{1}{a^2}x^2+\frac{1}{b^2}y^2=1.\)
  4. Evaluate \(\oint y^2\,\mathrm{d}x + 3xy\,\mathrm{d}y,\) where \(C\) is the boundary of the semiannular region \(R\) in the upper half-plane between the circles \(x^2+y^2=1\) and \(x^2+y^2=4.\)
  5. For the vector field \(\bm{F}\) defined as
    \(\displaystyle \bm{F}(x,y) = \frac{-y}{x^2+y^2}\mathbf{i}+\frac{x}{x^2+y^2}\mathbf{j}\)
    show that \(\int_C \bm{F}\cdot\mathrm{d}\bm{r} = 2\pi\) for every positively oriented simple closed path that encloses the origin.
  6. For \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\) compute the curl of \(\bm{F}.\)
  7. Show that the vector field \(\bm{F}\) defined as \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\) is not conservative.
  8. Show that the vector field \(\bm{F}\) defined as \(\bm{F}(x,y,z) = y^2 z^3\mathbf{i} + 2xyz^3 \mathbf{j} +3xy^2z^2\mathbf{k}\) is a conservative vector field, and find an example of a function \(f\) such that \(\bm{F} = \nabla f.\)
  9. For \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\) compute the divergence of \(\bm{F}.\)
  10. Show that the vector field \(\bm{F}\) defined as \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\) cannot be written as the curl of another vector field. I.e. there is no vector field \(\bm{G}\) such that \(\bm{F} = \operatorname{curl}\bm{G}.\)
  11. Evaluate \(\int_C \bm{F}\cdot\mathrm{d}\bm{r},\) where \(\bm{F}(x,y,z) = -y^2\mathbf{i} +x\mathbf{j}+z^2\mathbf{k}\) and \(C\) is the curve of intersection of the plane \(y+z=2\) and the cylinder \(x^2+y^2=1.\) (Orient \(C\) to be counterclockwise when viewed from above.)
  12. Use Stokes’ Theorem to compute the integral \(\iint_S \operatorname{curl}\bm{F}\cdot\mathrm{d}\bm{S},\) where \(\bm{F}(x,y,z) = xz\mathbf{i}+yz\mathbf{j}+xy\mathbf{k}\) and \(S\) is the part of the sphere \(x^2+y^2+z^2=4\) that lies inside the cylinder \(x^2+y^2=1\) and above the \(xy\)-plane.
  13. Calculate the flux of the vector field \(\bm{F}(x,y,z) = z\mathbf{i}+y\mathbf{j}+x\mathbf{k}\) across the unit sphere \(x^2+y^2+z^2=1.\)
  14. Evaluate \(\iint_S \bm{F}\cdot\mathrm{d}\bm{S},\) where \[ \bm{F}(x,y,z) = xy\mathbf{i} + \Big(y^2+\mathrm{e}^{xz^2}\Big)\mathbf{j} + \sin(xy)\mathbf{k} \] and \(S\) is the surface of the region \(E\) bounded by the parabolic cylinder \(z=1-x^2\) and the planes \(z=0\) and \(y=0\) and \(y+z=2.\)

Problems & Challenges

  1. Schey Let \(\mathbf{i}\) and \(\mathbf{j}\) and \(\mathbf{k}\) be the unit basis vectors of the standard rectangular coordinate system on \(\mathbf{R}^3,\) and let \(\mathbf{e}_r\) and \(\mathbf{e}_\theta\) and \(\mathbf{e}_z\) be the unit basis vectors of the standard cylindrical coordinate system on \(\mathbf{R}^3.\)
    1. Show that \[\begin{align*} \mathbf{i} &= \mathbf{e}_r\cos(\theta) - \mathbf{e}_\theta\sin(\theta) \\\mathbf{j} &= \mathbf{e}_r\sin(\theta) + \mathbf{e}_\theta\cos(\theta) \\\mathbf{k} &= \mathbf{e}_z\,. \end{align*}\]
    2. For a vector field \(\bm{F}(r,\theta,z) = P\mathbf{e}_r + Q\mathbf{e}_\theta + R\mathbf{e}_z,\) show that \(\operatorname{div}\bm{F}\) is given by \[\operatorname{div}\bm{F} = \frac{1}{r}\frac{\partial}{\partial r}\big(rP\big) + \frac{1}{r}\frac{\partial Q}{\partial \theta} + \frac{1}{r}\frac{\partial R}{\partial z}\,. \]
    3. Determine an analogous formula for \(\operatorname{curl}\bm{F}\) in cylindrical coordinates.
  2. Schey Let \(\mathbf{i}\) and \(\mathbf{j}\) and \(\mathbf{k}\) be the unit basis vectors of the standard rectangular coordinate system on \(\mathbf{R}^3,\) and let \(\mathbf{e}_\rho\) and \(\mathbf{e}_\theta\) and \(\mathbf{e}_\varphi\) be the unit basis vectors of the standard spherical coordinate system on \(\mathbf{R}^3.\)
    1. Show that \[\begin{align*} \mathbf{i} &= \mathbf{e}_\rho\sin(\theta)\cos(\varphi) + \mathbf{e}_\theta\cos(\theta)\cos(\varphi) - \mathbf{e}_\varphi\sin(\varphi) \\\mathbf{j} &= \mathbf{e}_\rho\sin(\theta)\sin(\varphi) + \mathbf{e}_\theta\cos(\theta)\sin(\varphi) - \mathbf{e}_\varphi\cos(\varphi) \\\mathbf{k} &= \mathbf{e}_\rho\cos(\theta) - \mathbf{e}_\theta\sin(\theta) \end{align*}\]
    2. For a vector field \(\bm{F}(\rho,\theta,\varphi) = P\mathbf{e}_\rho + Q\mathbf{e}_\theta + R\mathbf{e}_\varphi,\) show that \(\operatorname{div}\bm{F}\) is given by \[ \operatorname{div}\bm{F} = \frac{1}{\rho^2}\frac{\partial}{\partial \rho}\big(r^2P\big) + \frac{1}{\rho\sin(\theta)}\frac{\partial}{\partial \theta}\big(\sin(\theta)Q\big) + \frac{1}{\rho\sin(\theta)}\frac{\partial R}{\partial \varphi} \]
    3. Determine an analogous formula for \(\operatorname{curl}\bm{F}\) in spherical coordinates.
  3. Schey For an arbitrary vector \(\bm{v}\) and \(\bm{r} = \langle x,y,z\rangle,\) show that \[ \operatorname{curl}\Bigl(\frac{\bm{v}\times\bm{r}}{2}\Bigr) = \bm{v}\,. \]
  4. Schey Verify Stokes’ Theorem for these surfaces \(S\) and vector fields \(\bm{F}.\)

    1. The surface \(S\) consisting of the five faces of the unit cube in the first quadrant that don’t lie in the \(xz\) plane, and \(\bm{F} = \langle z^2, -y^2, 0\rangle.\)
    2. The surface that is the eighth of the unit sphere centered at the origin that lies entirely inside the first quadrant, and \(\bm{F} = \langle y,z,x \rangle.\)
  5. Stewart Prove the following identity: \[ \nabla\bigl(\bm{F}\cdot\bm{G}\bigr) = \bigl(\bm{F}\cdot\nabla\bigr) \bm{G} + \bigl(\bm{G}\cdot\nabla\bigr) \bm{F} + \bm{F} \times \operatorname{curl}\bm{G} + \bm{G} \times \operatorname{curl}\bm{F} \]
  6. Schey Recalling that \(\operatorname{div} = (\nabla\cdot)\) and that \(\operatorname{grad} = (\nabla)\) and that \(\operatorname{curl} = (\nabla\times)\) and that \(\nabla^2\) is the Laplace operator, verify the following identities concerning arbitrary differentiable scalar functions \(f\) and \(g\) and arbitrary differentiable vector fields \(\bm{F}\) and \(\bm{G}.\)

    \( \nabla(fg) = f\nabla g + g\nabla f \)
    \( \nabla\cdot \bigl(f\bm{F}\bigr) = f\nabla\cdot\bm{F} + \bm{F}\cdot\nabla f\)
    \( \nabla\cdot\bigl(\bm{F}\times\bm{G}\bigr) = \bm{G} \cdot \bigl(\nabla\times\bm{F}\bigr) - \bm{F} \cdot \bigl(\nabla\times\bm{G}\bigr) \)
    \( \nabla\times \bigl(f\bm{F}\bigr) = f\nabla\times\bm{F} - \bm{F}\times\nabla f\)
    \( \nabla\times\bigl(\bm{F}\times\bm{G}\bigr) = \bigl(\bm{G}\cdot\nabla\bigr)\bm{F} - \bigl(\bm{F}\cdot\nabla\bigr)\bm{G} + \bm{F}\bigl(\nabla\cdot\bm{G}\bigr) - \bm{G}\bigl(\nabla\cdot\bm{F}\bigr) \)
    \( \nabla\times\bigl(\nabla\times\bm{F}\bigr) = \nabla\bigl(\nabla\cdot\bm{F}\bigr) - \nabla^2\bm{F}\)
  7. Stewart Find the positively oriented simple closed curve \(C\) for which the value of the line integral is a maximum. \[ \oint\limits_C \bigl(y^3-y\bigr)\,\mathrm{d}x - \bigl(2x^3\bigr)\,\mathrm{d}y \]