Question

State the Divergence Theorem.

Short answer

The Divergence Theorem states: Let \(E\) be a solid region in \(R^3\) and let \(S\) be the boundary surface of \(E\). Suppose that \(F\) is a vector field whose component functions have continuous partial derivatives on an open region containing \(E\), then \[\int_ \int_{S} F \cdot dS = \int_ \int_ \int_E \nabla \cdot F dV\] Here, the surface integral measures the total flow of \(F\) across \(S\), while the volume integral measures the divergence of \(F\) within \(E\).

Step-by-step solution

State the theorem

The Divergence Theorem can be stated as follows: Let \(E\) be a solid region in \(R^3\) and let \(S\) be the boundary surface of \(E\). Suppose that \(F\) is a vector field whose component functions have continuous partial derivatives on an open region containing \(E\), then \[\int_ \int_{S} F \cdot dS = \int_ \int_ \int_E \nabla \cdot F dV\]

Explain the components of the theorem

In this theorem, \(S\) refers to the boundary surface of the solid \(E\). \(F\) is a vector field. The dot product \(F \cdot dS\) measures the flow of \(F\) across a small piece of the surface \(S\). \(\nabla \cdot F\) denotes the divergence of \(F\) and \(dV\) represents a small volume element in \(E\). The notation \(\int_ \int_{S}\) represents a surface integral over \(S\), while \(\int_ \int_ \int_E\) represents a triple integral over the volume of \(E\).