11.2: Divergence and Curl
- Page ID
- 144362
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- Find the divergence and the curl of \(\mathbf{F}(x, y, z) = xy^2z^4\,\mathbf{i} + (2x^2y + z)\,\mathbf{j} + y^3z^2\,\mathbf{k}\).
- Find the divergence and the curl of \(\mathbf{F}(x, y, z) = (x^2z, y^2x, y + 2z)\).
- Find the divergence and the curl of \(\mathbf{F}(x, y, z) = 3xyz^2\,\mathbf{i} + y^2\sin z\,\mathbf{j} + y^3z^2\,\mathbf{k}\).
- Find the divergence and the curl of \(\mathbf{F}(x, y, z) = (x^2yz, xy^2z, xyz^2)\) at \((x, y, z) = (2, 1, -1)\).
- Find the divergence and the scalar curl of \(\mathbf{F}(x, y) = (-y, x)\).
- Find the divergence and the scalar curl of \(\mathbf{F}(x, y) = \sin x\,\mathbf{i} + \cos y\,\mathbf{j}\).
- Find the divergence and the scalar curl of \(\mathbf{F}(x, y) = \dfrac{1}{\sqrt{x^2 + y^2}}(x, y)\) at \((x, y) = (1, -1)\).
- Let \(\mathbf{r} = (x, y, z)\) and \(r = \norm{\mathbf{r}}\). Find the divergence and curl of \(\dfrac{\mathbf{r}}{r}\).
- Let \(f: \mathbb{R}^3 \to \mathbb{R}\) have continuous second-order partial derivatives. Show that \(\nabla \times (\nabla f) = \mathbf{0}\).
- Let the component functions of \(\mathbf{F}: \mathbb{R}^3 \to \mathbb{R}^3\) have continuous second-order partial derivatives. Show that \(\nabla \cdot (\nabla \times \mathbf{F}) = 0\).