In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point. The divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For questions about divergent sequences use [tag:convergence-divergence]
Let $\vec{F}:\mathbb{R}^n\to\mathbb{R}^n = \langle F_1(x_1,\ldots,x_n),\ldots,F_n(x_1,\ldots,x_n)\rangle$ be a vector field. The divergence of $\vec{F}$ is the following: $$ \text{div}(\vec{F}) = \sum_{k=1}^{n} \frac{\partial}{\partial x_k}F_k $$Note that this is a scalar, not a vector (compare to the gradient). By a slight abuse of notation, if $\nabla$ is the gradient operator, we may write $$ \text{div}(\vec{F}) = \nabla \cdot \vec{F} $$The divergence theorem is a powerful result in vector calculus. Roughly speaking, it states that if $V$ is a surface and $\partial V$ is its boundary, and if $\vec{F}$ is a continuously differentiable vector field, then $$ \iint_{\partial V} \vec{F}\cdot dS = \iiint_{V} \text{div}(\vec{F})\,dV $$ The RHS is often easier to calculate as the divergence of $\vec{F}$ is often much simpler to work with; additionally, the boundary $\partial V$ often requires different parametrizations whereas $V$ can sometimes be described as a single iterated integral.
