1. TK

2. James Stewart

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

3. James Stewart

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