Vector Calculus Problems

Trivium

  1. Given a vector field over either \(\mathbf{R}^2\) or \(\mathbf{R}^3,\) determine whether or not it’s conservative. If it’s conservative, determine its potential function.
  2. Given a vector field over either \(\mathbf{R}^2\) or \(\mathbf{R}^3,\) compute its divergence and its curl.
  3. Explain what divergence and curl are, and express some intuition on what they compute. Concretely, given a vector field over either \(\mathbf{R}^2\) or \(\mathbf{R}^3\) and a region with a boundary in that space, determine based on intuition whether the divergence or curl of the vector field over that region is zero.
  4. Compute the value of a line integral over a piecewise-smooth curve in a vector field. If the curve is closed, compute its value either directly or as an integral over its interior via either Green’s Theorem or Stokes’ Theorem.
  5. Compute the value of a surface integral over a piecewise smooth surface in a vector field. If the surface is closed, compute its value either directly or as an integral over its interior via either Green’s Theorem or the Divergence Theorem.

“Exercises” from Stewart that Look Like They Could Be the Basis for a Reasonable Exam Question

Problems & Challenges

  1. Schey

    Instead of using arrows to represent vector functions, we can use families of curves called field lines. In two dimensions, a curve \(\bigl(x(t), y(t)\bigr)\) is a field line of the vector field \(\bm{F}\) if at each point \((x_0, y_0)\) on the curve, \(\bm{F}(x_0,y_0)\) is tangent to the curve.

    1. Realize that the field lines \(\bigl(x(t), y(t)\bigr)\) for a vector field \(\bm{F} = P\mathbf{i} + Q\mathbf{j}\) are solutions to the differential equation \[ \frac{\mathrm{d}y/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t} = \frac{Q}{P}\,. \]
    2. Determine the family of field lines for each of these vector fields.
      \(\bm{F} = \bigl\langle y, x \bigr\rangle\)
      \(\bm{F} = \bigl\langle x, -y \bigr\rangle\)
      \(\bm{F} = \bigl\langle 0, x \bigr\rangle\)
      \(\bm{F} = \bigl\langle y, xy \bigr\rangle\)
      \(\bm{F} = \bigl\langle 1, y \bigr\rangle\)
      \(\displaystyle \bm{F} = \biggl\langle \frac{y}{\sqrt{x^2+y^2}}, \frac{x}{\sqrt{x^2+y^2}} \biggr\rangle\)

  2. Schey

    Sometimes the value of a surface integral can be reasoned out without working through the usual arduous procedure simply by remembering that the surface integral calculates the flux, the “sum of flow”, across the surface. For each of the following vector fields \(\bm{F}\) and surfaces \(S,\) reason out the value of the surface integral \(\iint_S \bm{F}\cdot\mathrm{d}\bm{S}.\)

    1. Consider the cube of side-length \(\ell\) in the first quadrant with diagonally opposite corners located at \((0,0,0)\) and \((\ell,\ell,\ell).\) The surface \(S\) consists of the three sides of this cube that lie in the coordinate planes, and \(\bm{F}(x,y,z) = \langle x,y,z \rangle.\)
    2. The surface \(S\) is the cylinder with height \(h\) and base of radius \(R\) in the \(xy\)-plane, and \(\bm{F}(x,y,z) = \bigl\langle x\ln(x^2+y^2), y\ln(x^2+y^2), 0 \bigr\rangle.\)
    3. The surface \(S\) is the sphere of radius \(R\) centered at the origin, and \[\bm{F}(x,y,z) = \biggl\langle x\mathrm{e}^{-\bigl(x^2+y^2+z^2\bigr)}, y\mathrm{e}^{-\bigl(x^2+y^2+z^2\bigr)}, z\mathrm{e}^{-\bigl(x^2+y^2+z^2\bigr)} \biggr\rangle.\]
    4. The surface \(S\) is the cube of side-length \(\ell\) in the first quadrant with diagonally opposite corners located at \((0,0,0)\) and \((\ell,\ell,\ell),\) and \(\bm{F}(x,y,z) = \langle f(x), 0, 0 \rangle\) where \(f\) is an abritrary scalar function of \(x\).

  3. 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.

  4. 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.

  5. Schey

    For an arbitrary vector \(\bm{v}\) and \(\bm{r} = \langle x,y,z\rangle,\) show that \[ \operatorname{curl}\Bigl(\frac{\bm{v}\times\bm{r}}{2}\Bigr) = \bm{v}\,. \]

  6. 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 \(\bm{F} = \langle y,z,x \rangle.\)

  7. Stewart

    Prove the following identity: \[ \nabla\bigl(\bm{F}\cdot\bm{G}\bigr) = \bigl(\bm{F}\cdot\nabla\bigr) \bm{G} + \bigl(\bm{G}\cdot\nabla\bigr) \bm{F} + \bm{F} \times \operatorname{curl}\bm{G} + \bm{G} \times \operatorname{curl}\bm{F} \]

  8. 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 \(\bm{G}.\)

    \( \nabla(fg) = f\nabla g + g\nabla f \)
    \( \nabla\cdot \bigl(f\bm{F}\bigr) = f\nabla\cdot\bm{F} + \bm{F}\cdot\nabla f\)
    \( \nabla\cdot\bigl(\bm{F}\times\bm{G}\bigr) = \bm{G} \cdot \bigl(\nabla\times\bm{F}\bigr) - \bm{F} \cdot \bigl(\nabla\times\bm{G}\bigr) \)
    \( \nabla\times \bigl(f\bm{F}\bigr) = f\nabla\times\bm{F} - \bm{F}\times\nabla f\)
    \( \nabla\times\bigl(\bm{F}\times\bm{G}\bigr) = \bigl(\bm{G}\cdot\nabla\bigr)\bm{F} - \bigl(\bm{F}\cdot\nabla\bigr)\bm{G} + \bm{F}\bigl(\nabla\cdot\bm{G}\bigr) - \bm{G}\bigl(\nabla\cdot\bm{F}\bigr) \)
    \( \nabla\times\bigl(\nabla\times\bm{F}\bigr) = \nabla\bigl(\nabla\cdot\bm{F}\bigr) - \nabla^2\bm{F}\)

  9. Stewart

    Find the positively oriented simple closed curve \(C\) for which the value of the line integral is a maximum. \[ \oint\limits_C \bigl(y^3-y\bigr)\,\mathrm{d}x - \bigl(2x^3\bigr)\,\mathrm{d}y \]