Proof of identities of divergence of vector fields

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You could just expand both sides and see whether they are equal. For example (i),

$$\nabla \cdot (\phi F)=\nabla\cdot (\phi f_1, \phi f_2, \phi f_3)=\frac{\partial (\phi f_1)}{\partial x}+\frac{\partial (\phi f_2)}{\partial y}+\frac{\partial (\phi f_3)}{\partial z}$$

On the other hand,

$$\nabla\phi \cdot F+\phi(\nabla \cdot F)=\left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z}\right)\cdot (f_1, f_2, f_3)+\phi \left(\frac{\partial f_1}{\partial x}+\frac{\partial f_2}{\partial y}+\frac{\partial f_3}{\partial z}\right)$$

You can use product rule to see they are equal. (ii) can be done similarly.

The additional ones are also similar, for example (ii),

$$\nabla \times (\phi F)=\left(\frac{\partial (\phi f_3)}{\partial y}-\frac{\partial (\phi f_2)}{\partial z}, \frac{\partial (\phi f_1)}{\partial z}-\frac{\partial (\phi f_3)}{\partial x}, \frac{\partial (\phi f_2)}{\partial x}-\frac{\partial (\phi f_1)}{\partial y}\right)$$,

and the right hand side,

$$\nabla \phi \times F +\phi (\nabla \times F)=\left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z}\right) \times F + \left(\phi \left(\frac{\partial f_3}{\partial y}-\frac{\partial f_2}{\partial z}\right),\, \phi \left(\frac{\partial f_3}{\partial x}-\frac{\partial f_1}{\partial z}\right),\, \phi \left(\frac{\partial f_2}{\partial x}-\frac{\partial f_1}{\partial y}\right)\right)$$

Again, using product rule and definition of curl, you can see they are the same.

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Updated on August 01, 2022

Comments

  • user189013
    user189013 over 1 year

    I want to prove some identities but I don't know how to do this.

    First of all,

    $φ : R^3 → R$ and vector fields $F = (f_1, f_2, f_3), G = (g_1, g_2, g_3) : R^3 → R^3$

    the two identities are:

    (i)$ ∇ · (φF) = ∇φ · F + φ(∇ · F) $ (ii) $∇ · (F × G) = G · (∇ × F) − F · (∇ × G)$

    Additional identities to prove:

    continuously differentiable scalar fields $φ, ψ : R^3 → R$ and vector field $F : R^3 → R^3$:

    (i) $∇(φ ψ) = φ∇ψ + ψ∇φ$ (ii) $∇ × (φF) = ∇φ × F + φ(∇ × F)$

  • user189013
    user189013 almost 9 years
    I have added some additional identities I want to prove if you can give me any hint.