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Chapter 13: Problem 19

State the Divergence Theorem.

Short Answer

Expert verified
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

01

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 \[\int_ \int_{S} F \cdot dS = \int_ \int_ \int_E \nabla \cdot F dV\]
02

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 \(E\).

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