Question

Expand the vector expressions in 3D and prove the following: (a) $(\mathbf{u} \cdot \nabla)\mathbf{u} = \nabla(\frac{1}{2}\mathbf{u}\cdot\mathbf{u}) - \mathbf{u} \times (\nabla \times \mathbf{u})$ (b) $\nabla \times \nabla \phi = 0$ (c) $\nabla \times (\mathbf{u} \times \mathbf{\omega}) = \mathbf{u} (\nabla \cdot \mathbf{\omega}) - \mathbf{\omega} (\nabla \cdot \mathbf{u}) - (\mathbf{u} \cdot \nabla)\mathbf{\omega} + (\mathbf{\omega} \cdot \nabla)\mathbf{u}$ (d) $\nabla \times \frac{\partial \mathbf{u}}{\partial t} = \frac{\partial}{\partial t}(\nabla \times \mathbf{u})$ (e) $\nabla \times \nabla^2 \mathbf{u} = \nabla^2 (\nabla \times \mathbf{u})$ Here $\mathbf{u}$ and $\mathbf{\omega}$ are 3D vectors and $\phi$ is a scaler.

          Expand the vector expressions in 3D and prove the following:
(a) $(\mathbf{u} \cdot \nabla)\mathbf{u} = \nabla(\frac{1}{2}\mathbf{u}\cdot\mathbf{u}) - \mathbf{u} \times (\nabla \times \mathbf{u})$
(b) $\nabla \times \nabla \phi = 0$
(c) $\nabla \times (\mathbf{u} \times \mathbf{\omega}) = \mathbf{u} (\nabla \cdot \mathbf{\omega}) - \mathbf{\omega} (\nabla \cdot \mathbf{u}) - (\mathbf{u} \cdot \nabla)\mathbf{\omega} + (\mathbf{\omega} \cdot \nabla)\mathbf{u}$
(d) $\nabla \times \frac{\partial \mathbf{u}}{\partial t} = \frac{\partial}{\partial t}(\nabla \times \mathbf{u})$
(e) $\nabla \times \nabla^2 \mathbf{u} = \nabla^2 (\nabla \times \mathbf{u})$
Here $\mathbf{u}$ and $\mathbf{\omega}$ are 3D vectors and $\phi$ is a scaler.

Expand the vector expressions in 3D and prove the following:
(a) (𝐮·∇)𝐮 = ∇((1)/(2)𝐮·𝐮) - 𝐮× (∇×𝐮)
(b) ∇×∇ϕ = 0
(c) ∇× (𝐮×ω) = 𝐮 (∇·ω) - ω (∇·𝐮) - (𝐮·∇)ω + (ω·∇)𝐮
(d) ∇×(∂𝐮)/(∂ t) = (∂)/(∂ t)(∇×𝐮)
(e) ∇×∇^2 𝐮 = ∇^2 (∇×𝐮)
Here 𝐮 and ω are 3D vectors and ϕ is a scaler.

Added by Marc A.

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