Calculus Small Changes, Big Ideas

Section 20.1 Conservative vector fields

• A vector field is called conservative if its line integral is path independent; that is, the choice of path does not change the value of the line integral. Under what conditions is a vector field conservative?

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• Theorem 11. Gradient thoerem.

  Suppose \(C\) is a smooth curve parametrized by the vector function \(\bv r(t)\) with \(a\le t\le b\text{,}\) and let \(f\) be a differentiable function whose gradient \(\nabla f\) is continuous. Then

  \begin{equation*} \int_{C}\nabla f\cdot\dd{\bv{r}}=f(\bv{r}(b))-f(\bv{r}(a)) \end{equation*}

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• If \(\bv F\) is a gradient field, it must be conservative.

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• As for the converse, if \(\bv F\) is conservative and is continuous on an open connected region \(D\text{,}\) then \(\bv F\) is a gradient field on \(D\text{.}\)

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• A vector field is conservative if and only if every closed path integral is zero.

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• The fact that the curl of a gradient is zero gives another test for whether a field is conservative. Suppose \(\bv{F}=(P,Q)\) is defined on an open simply connected region \(D\) and has continuous first order partial derivatives.

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  If \(\DS\pdv{P}{y}=\DS\pdv{Q}{x}\) throughout \(D\text{,}\) then \(\bv{F}\) is conservative.

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• Make sure to do an example using energy to show the conservation law actually happening!

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