6.1 Vector fields  (Page 7/15)

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Conservative vector fields also have a special property called the cross-partial property . This property helps test whether a given vector field is conservative.

Proof

Since F is conservative, there is a function $f(x,y)$ such that $\text{∇}f=\text{F}.$ Therefore, by the definition of the gradient, $f_x=P$ and $f_y=Q.$ By Clairaut’s theorem, $f_{xy}=f_{yx},$ But, $f_{xy}=P_y$ and $f_{yx}=Q_x,$ and thus $P_y=Q_x.$

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Clairaut’s theorem gives a fast proof of the cross-partial property of conservative vector fields in ${ℝ}^3,$ just as it did for vector fields in ${ℝ}^2.$

[link] shows that most vector fields are not conservative. The cross-partial property is difficult to satisfy in general, so most vector fields won’t have equal cross-partials.

We conclude this section with a word of warning: [link] says that if F is conservative, then F has the cross-partial property. The theorem does not say that, if F has the cross-partial property, then F is conservative (the converse of an implication is not logically equivalent to the original implication). In other words, [link] can only help determine that a field is not conservative; it does not let you conclude that a vector field is conservative. For example, consider vector field $\text{F}(x,y)=⟨x^{2}y,\frac{x^3}{3}⟩.$ This field has the cross-partial property, so it is natural to try to use [link] to conclude this vector field is conservative. However, this is a misapplication of the theorem. We learn later how to conclude that F is conservative.

Key concepts

• A vector field assigns a vector $\text{F}(x,y)$ to each point $(x,y)$ in a subset D of ${ℝ}^2\text{or}{ℝ}^3.$ $\text{F}(x,y,z)$ to each point $(x,y,z)$ in a subset D of ${ℝ}^3.$
• Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography.
• We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern.
• A vector field $\text{F}$ is called conservative if there exists a scalar function $f$ such that $\text{∇}f=\text{F}.$