I have been all over the internet, but I just can't make sense of this stuff. I have done my best to learn from my textbook and different websites, but this is confusing for me. I haven't taken any calculus in years, and I'm jumping in headfirst. If anyone can help me understand how to prove these using Cartesian Tensor Notation, I would really appreciate it!
First identity: $\nabla \times (\nabla \times a) = \nabla \cdot (\nabla \cdot a) - \nabla^2a$
Second identity: $\nabla \cdot (ab) = a \cdot \nabla b +b(\nabla \cdot a)$
Third identity: $\nabla \cdot (\delta f) = \nabla f$
Fourth identity: $\delta : \nabla a = \nabla \cdot a $
Where $\delta f$ represents a small change in $f$.
Thanks everyone
Recall the following: $$ \text{grad($\it f$)} = \nabla f = \frac{\partial f}{\partial x}\hat i + \frac{\partial f}{\partial y}\hat j + \frac{\partial f}{\partial z}\hat k \\ \nabla \cdot f = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z} \\ \nabla \times f = \begin{vmatrix} \hat i & \hat j & \hat k \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z}\end{vmatrix}$$
I will prove the second identity as an example, and the others will follow via the same methods.
$\text{Using the first two formulae: }\nabla \cdot (ab) = \frac{\partial (ab)}{\partial x} + \frac{\partial (ab)}{\partial y} + \frac{\partial (ab)}{\partial z} = \text{(by the product rule) } (\frac{\partial a}{\partial x} b + a\frac{\partial b}{\partial x}) + (\frac{\partial a}{\partial y} b + a\frac{\partial b}{\partial y}) + (\frac{\partial a}{\partial z} b + a\frac{\partial b}{\partial z}) = a(\frac{\partial (b)}{\partial x} + \frac{\partial (b)}{\partial y} + \frac{\partial (b)}{\partial z}) + b(\frac{\partial (a)}{\partial x} + \frac{\partial (a)}{\partial y} + \frac{\partial (a)}{\partial z}) = a \cdot (\nabla b) + b \ (\nabla \cdot a) \space \blacksquare$
The others would be good exercise, if you are stuck leave a comment and I will edit my answer to include hints.