Divergence and curl
Problem 12.31 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle x^{2}, y^{2}, z^{2} \rangle \).
- \[ \frac{d}{d x} x^{2} + \frac{d}{d y} y^{2} + \frac{d}{d z} z^{2} = 2 x + 2 y + 2 z \]div F.✓ Proved
- \[ \left[\begin{matrix}- \frac{d}{d z} y^{2} + \frac{d}{d y} z^{2}\\\frac{d}{d z} x^{2} - \frac{d}{d x} z^{2}\\- \frac{d}{d y} x^{2} + \frac{d}{d x} y^{2}\end{matrix}\right] = \left[\begin{matrix}0\\0\\0\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = 2 x + 2 y + 2 z,\quad \nabla\times\mathbf{F} = \langle 0, 0, 0 \rangle \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | difference quotients of each component at (0.3, 0.7, 1.1) agree |
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/divergence_curl, checked 2026-09-26 with SymPy 1.14.0.