# Physical meaning of divergence

While reading the section on Hamiltonian mechanics in Taylor's Classical mechanics, I realized that I didn't fully understand what he was saying when he was explaining why $$\nabla\cdot\vec{F(\vec{x_0})}=\frac{1}{V}\frac{dV}{dt},$$for small volumes around $\vec{x_0}.$

$$\frac{dV}{dt}=\int_{\partial V}\vec{n}\cdot\vec{F}dA=\int_V\nabla\cdot\vec{F}dV.$$ Taking over a small enough volume, $\nabla\cdot\vec{F}$ is constant, factors out of the integral, and the result follows.

I'm not sure why $\nabla\cdot\vec{F}$ is locally constant. The first time I read this, I thought is was just a matter of assuming sufficient smoothness on the partial derivatives, but after thinking for a second, that can't imply locally constant. So I'm a little confused.

• Let $U$ be a simply connected open subset of $\mathbb R^3$. Take any smooth function $F : U \to \mathbb R$. The author claims that as the volume $V_U$ of $U$ approach 0, the following approximate result holds: $$\int_U F dU \simeq V_U F$$ where $dU$ is the pullback of the usual $\mathbb R^3$ volume form by the inclusion. See if you can convince yourself of that.
– zzz
Jun 26, 2015 at 23:46

Think about it one more time. If $\vec{F}$ has continuous partial derivatives, then $$\vec\nabla\cdot\vec{F}=\sum_i \frac{\partial F_i}{\partial x_i}$$ is also continuous. If a function is continuous, it's approximately constant on sufficiently small volumes: that's pretty much the definition of continuity! So your original understanding was just fine.
Suppose you have a volume $v$ in vector field $\mathbf{h}$. Divide it into two parts: $$v = v_1 +v _2$$. Now the flux out of volume $v$ is given by : $\int \mathbf{h}\cdot da_v = \int \mathbf{h} \cdot da_{v_1} + \int \mathbf{h}\cdot da_{v_2}$. Start dividing the volume into $N$ parts so that $v = \sum_{i=1}^N v_i$. So, the net flux out of volume $v$ is $$\int \mathbf{h} \cdot da_v= \sum_{i=1}^N \int \mathbf{h} \cdot da_{v_i}$$. This is a macrosopic quantity. However, we want to find some microscopic property; in order to do this, we make $N \to \infty$ so that $da_{v_i} \to 0$. Let the flux out of such infinitesimal volume $v_i$ is $\int\mathbf{ h}\cdot da_i$. This quantity is surely approximating to $0$. But if we take the ratio of the flux divided by the volume it encloses, we can get a finite quantity. This is what we call divergence. $$\text{div} \mathbf{h}_i \equiv \lim_{v_i \to 0} \dfrac{1}{v_i} \int \mathbf{h}\cdot da_{v_i}$$.