Div, grad & curl

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Let $F(x,y,z) = \left(5y-6z,\ 4z+6x,\ -5x-4y\right) = (P, Q, R)$. Find the third ($z$-)component of the curl $\nabla\times F$, given by $\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$.

2. Warm-up

   Let $F(x,y,z) = \left(6y+3z,\ -6z-2x,\ 6x-6y\right) = (P, Q, R)$. Find the third ($z$-)component of the curl $\nabla\times F$, given by $\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$.

3. Warm-up

   Let $F(x,y,z) = \left(3y-5z,\ 5z+4x,\ -3x+4y\right) = (P, Q, R)$. Find the third ($z$-)component of the curl $\nabla\times F$, given by $\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$.

Problems shown under their original terms: Generated (template, solver-computed).