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

• 12–24, 41–43
• 12–24, 29, 35! 41!
• 5–18, 21–23
• 14! 21–34
• 5–32
• 2–14, 20, 21
• 1–17

Problems & Challenges

1. James Stewart

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

2. James Stewart

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