Divergence Theorem

Section 5.7 Divergence Theorem

On one side of Green’s Theorem is the vector line integral along a curve. It measures how much the field is in the same direction as the curve. The divergence theorem on one side has a vector line integral calculating the component of the field normal to the curve. This is sometimes called flux and is calculated as follows

\begin{align} \int_C \vec{F} \cdot \vec{n} \; ds & = \int_C \vec{F} \cdot \frac{\langle g'(t),-f'(t) \rangle}{\|\vec{r}^\prime(t) \|} \| \vec{r}^\prime \| \; dt\tag{5.1}\\ & = \int_C \langle M,N \rangle \cdot \langle g^\prime(t),-f^\prime(t) \rangle \; dt\notag\\ & = \int_C M g^\prime(t) \; dt - \int_C N f^\prime(t) \; dt\notag \end{align}

The normal vector is used to determine the direction (across the curve) so it’s magnitude must be removed (normalized). The integral is with respect to the length of the curve, so the magnitude of the curve is re-introduced. As such, the final result is a dot product with \(\vec{n}\) rather than \(\hat{n}\) as shown in (5.1).

Theorem 5.7.1. Divergence.

For a simple, closed region \(D\) inside a curve \(C\) parameterized so that \(D\) is on the left of \(C\) and a vector field \(\vec{F}=(M(x,y),N(x,y))\) whose derivatives are continuous in the region

\begin{equation*} \oint_C \vec{F} \cdot \vec{n} \; ds = \iint_D \text{div} \vec{F} \; dA \end{equation*}

Lessons for Multivariable Calculus

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Section 5.7 Divergence Theorem

Theorem 5.7.1. Divergence.