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.