I have some scalar field $u:D \rightarrow\mathbb R; \space \space D\subset \mathbb R^3$
and a vector field $\vec{v}: D\rightarrow \mathbb R^3$
and I want to show that:
curl$(u\vec{v)}=$grad$(u)\times \vec{v}+u\space$rot$(\vec{v})$
My question is: How do I multiply a vector field by a scalar field?
Can I write curl$(u\vec{v})$ like this:
curl$(\vec{v})=$curl$\begin{pmatrix}uv_1\\uv_2\\uv_3\end{pmatrix}=$
$\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\\frac{d}{dx}&\frac{d}{dy}&\frac{d}{dz}\\uv_1&uv_2&uv_3\end{vmatrix}=\hat{i}(\frac{d}{dy}(uv_3)-\frac{d}{dz}(uv_2))-\hat{j}(\frac{d}{dx}(uv_3)-\frac{d}{dz}(uv_1))+\hat{k}(\frac{d}{dx}(uv_2)-\frac{d}{dy}(uv_1))$
or am I doing something wrong?
Edit:
grad$(u)\times\vec{v}+u\space $curl$(\vec{v})=(\frac{du}{dx}\hat{i}+\frac{du}{dy}\hat{j}+\frac{du}{dz}\hat{k}) \times (v_1 \hat{i}+v_2\hat{j}+v_3\hat{k})+u\space $curl$(\vec{v})$
$=\hat{i}(\frac{du}{dy}(v_3)-\frac{du}{dz}(v_2))-\hat{j}(\frac{du}{dx}(v_3)-\frac{du}{dz}(v_1))+\hat{k}(\frac{du}{dx}(v_2)-\frac{du}{dy}(v_1))+u[\hat{i}(\frac{d}{dy}(v_3)-\frac{d}{dz}(v_2))-\hat{j}(\frac{d}{dx}(v_3)-\frac{d}{dz}(v_1))+\hat{k}(\frac{d}{dx}(v_2)-\frac{d}{dy}(v_1)]$
$\not=\hat{i}(\frac{d}{dy}(uv_3)-\frac{d}{dz}(uv_2))-\hat{j}(\frac{d}{dx}(uv_3)-\frac{d}{dz}(uv_1))+\hat{k}(\frac{d}{dx}(uv_2)-\frac{d}{dy}(uv_1))$
I guess I am making a mistake somewhere but I can't find it.
Edit 2:
grad$\space u \times \vec{v}=(\frac{du}{dx}\hat{i}+\frac{du}{dy}\hat{j}+\frac{du}{dz}\hat{k}) \times (v_1 \hat{i}+v_2\hat{j}+v_3\hat{k})$
$=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\\frac{du}{dx}&\frac{du}{dy}&\frac{du}{dz}\\v_1&v_2&v_3\end{vmatrix}=\hat{i}(\frac{du}{dy}(v_3)-\frac{du}{dz}(v_2))-\hat{j}(\frac{du}{dx}(v_3)-\frac{du}{dz}(v_1))+\hat{k}(\frac{du}{dx}(v_2)-\frac{du}{dy}(v_1))$
We will use the notation such that summations are implied over repeated indices. SO, for example, $\vec A =\sum_{i=1}^3 \hat x_iA_i$ will be abbreviated $\vec A =\hat x_iA_i$.
$$\begin{align} \nabla \times (u\vec v)&=\left(\hat x_i \frac{\partial}{\partial x_i}\right)\times(u\hat x_j v_j)\\\\ &=(\hat x_i \times \hat x_j)\frac{\partial (uv_j)}{\partial x_i}\\\\ &=(\hat x_i \times \hat x_j)\left(u\frac{\partial v_j}{\partial x_i}+v_j\frac{\partial u}{\partial x_i}\right)\\\\ &=u\left(\hat x_i \frac{\partial }{\partial x_i}\right)\times (\hat x_jv_j)+\left(\hat x_i \frac{\partial u}{\partial x_i}\right)\times (\hat x_jv_j)\\\\ &=u\nabla \times \vec v+\nabla u \times \vec v \end{align}$$