I must learn tons of identities for vector calculus for Electromagnetism, but they were taught as a list of items to memorize. Some of them are reasonable to follow (such as gradient or curl acting on sum or product of vectors), but I'm lost with curl and vector product.
I lack any formal math background, so I'd like a book/ page where these expressions are justified/ derived in an understandable way.
Thanks in advance!
Stewart's Calculus has an intro to vector calculus, but has some gaps. Learn the fundamental definitions of the vector derivatives and you can usually derive the identities you need from there.
By the chain rule, $df=\frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y} dy$. Note there are no explicit references to vectors yet. Take $\vec{ds}=dx\hat{i}+ dy \hat{j}$ and $\nabla f = \frac{\partial f}{\partial x}\hat {i} + \frac{\partial f}{\partial y} \hat{j}$
You can rewrite it as $df= \nabla f \cdot \vec{ds}$ where $\nabla f$ is the gradient of $f$.
$d(fg)=f dg + g df = f (\nabla g\cdot \vec{ds})+g(\nabla \vec{f} \cdot \vec{ds})=(f\nabla g + g \nabla f)\cdot \vec{ds}$
So $\nabla(fg)=f\nabla g + g \nabla f$. Properties of the partial derivativfe translate into properties of the gradient.
You can use something similar to prove $\nabla (f+g)= \nabla f + \nabla g$
You can also use the principle to derive the gradient in different coordinate systems.
For example, in cylindrical coordinates, the line element is $\vec{ds}=dr \hat{r} +r d\theta \hat{\theta} $.
We need $df = \nabla f \cdot \vec{ds}= \frac{\partial f}{\partial r}dr + \frac{\partial f}{\partial \theta} d \theta=(\nabla f )_rdr+(\nabla f)_\theta r d\theta$. Match the coefficients of dr on the right to those on the left, you get $(\nabla f)_r=\frac{\partial f}{\partial r}$ and $(\nabla f)_\theta r d\theta= \frac{\partial f}{\partial \theta}$. From this it follows that $\nabla f=\frac{\partial f}{\partial r} \hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \hat{\theta} $
Fundamentally the divergence is $\nabla \cdot \vec{E} = \lim_{dV\to 0} \frac{\sum_{k=1}^3 E_i \cdot \hat{n_i} dA}{dV}$ where $dV$ is differential volume element and $dA$ is differential surface area element perpendicular to the line element under consideration.
For Spherical Coordinates, here's the divergence
$\nabla \cdot \vec{E}= \lim_{dV \to 0} \frac{\int E_r r^2 \sin \phi d\phi d\theta+\int E_\theta r drd\phi+ \int E_\phi r (\sin \phi) dr d\theta}{\int r^2\sin \phi dr d\phi d\theta}$
Cancel out the differentials where applicable, then the ones that remain tell you what parital derivatives you need to us to get the divergence.
$\nabla \cdot \vec{E} = \lim_{dV \to 0} \frac{1}{r^2}\frac{\partial}{\partial r}(r^2E_r) + \frac{1}{r \sin \phi}\frac{\partial}{\partial \theta}(E_\theta) + \frac{1}{r \sin \phi}\frac{\partial (\sin\phi E_\phi)}{\partial \phi}$
If you want to find $\nabla \cdot (f\vec{E})$, just plug in $fE_i$ for $E_i$ in the above and you get $\nabla \cdot (f\vec{E})= \nabla f \cdot \vec{E}+f \nabla \cdot \vec{E}$.
There's an integral definition of the curl you can use in similar ways, but I find its easier to use linear properties of vectors to remember these. $\vec{E} = E_i \hat{e_i}+E_j \hat{e_j}+E_k \hat{e_k}$ (where $e_i$ aren't necessarily the Cartesian basis vectrs. )
$\nabla \times \vec{E} = \nabla(E_i) \times \hat{e_i} + E_i (\nabla \times \hat{e_i})+ \nabla(E_i)\times \hat{e_i} + E_j (\nabla \times \hat{e_j})+ \nabla(E_k)\times \hat{e_k} + E_k (\nabla \times \hat{e_k})$
You need to consider gradients of the components and curls of the basis vectors.
Then keep in mind that the curl of a gradient is 0, i.e. $\nabla \times \nabla f = 0$. For any Cartesian basis vector, $\hat{e_i}= \nabla (x_i)$, so $\nabla \times \hat{e_i} = \nabla \times \nabla (x_i) = 0$, so the curl of the basis vectors are all 0. This is not always the case.
$\hat {r}= \nabla(r)$, so $\nabla \times \hat{r}=0$. $\hat {k} =\nabla (z)$ so $\nabla \times \hat{k}=0 $
$\nabla \times \hat {\theta}$ is more complicated since there is no $V$ so that $\nabla V = \hat{\theta}$. You can derive it using the property $\nabla \times \nabla f=0$ or memorize the curl for the one basis vector that has non zero curl.
Finally, by definition $\nabla^2 f= \nabla \cdot (\nabla f) $.
After you know those fundamentals you can sort out more complicated expressions from the parts, e.g. $\nabla \cdot (\nabla \times \vec{E})=0$