# Tricks to remember Vector Calculus formulas

I bumped into this chart to remember some vector calculus identities: I was wondering if you know any other trick/chart or what have you to remember Vector Calculus Theorems (e.g. Divergence, Stokes, Greens, etc) and other vector calculus identities, for example $$\mathbb{\nabla}\cdot(\mathbf{F}\times \mathbf{G}) = \mathbf{G}\cdot(\mathbf{\nabla}\times\mathbf{F})-\mathbf{F}\cdot(\mathbf{\nabla}\times\mathbf{G})$$ Some tricks might be thinking of the product rule for the following, for instance. $$\mathbf{\nabla}\cdot(\phi\psi) = \psi\nabla\phi+\phi\nabla\psi$$ is there any other? For example in the one above this one, how would someone intuitively explain the minus sign? of course it comes out if someone computes the expression, but are there tricks to remember them better?

EDIT

In the graph we have C = curl, D = divergence, G = Gradient. The blue arrows means that if you apply the operator at the beginning, to the operator at the end, in the direction of the arrow, then you get what is written above. L stands for laplacian, so L scalar is the laplacian of a scalar field, whereas CC+Lvect stands for curl of curl and laplacian of a vector field (i.e. the gradient, if we have a scalar field)

• you should define all the terms in the graph. I guessed Divergence, Curl, Grad. Lscalar = Laplacian ? CC+Lvect is quite cryptic. – Fabrice NEYRET Dec 20 '16 at 16:49
• seeing the vote to close it, I don't think there are too many tricks, as shown below, there is one trick for all the basic theorems – Euler_Salter Dec 20 '16 at 17:05

## 1 Answer

The Stokes' theorem unifies the theorems regarding integrations you mentioned. There is really only one trick (which could be one's motivation to learn differential geometry):

$$\int_{\partial\Omega}\omega=\int_{\Omega}d\omega.$$

• Which is "just" the generalization of int_a,b{df(x)} = f(b)-f(a) in R. – Fabrice NEYRET Dec 20 '16 at 16:48
• Wow I had no idea. Okay, I'll have a look a the resources you posted in your answer and come back. Just as a curiosity, what field is differential geometry in? Meaning, do I need to know anything about groups, homologies or any pure mathematics to study it? – Euler_Salter Dec 20 '16 at 16:48
• @Euler_Salter: You might begin with Spivak's Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus. – Jack Dec 20 '16 at 16:52
• @Jack thank you! I'll have a look – Euler_Salter Dec 20 '16 at 16:58