Vector field

Conservative vector field

A conservative vector field¹ is a Vector field in which the line integral is path-independent, i.e. only depends on the start and end points. #m/def/anal/vec

Any conservative vector field may be expressed as the gradient field of some scalar field, called the scalar potential, such that

Properties and examples

• The Circulation around any closed path is 0
• The Curl of any conservative vector field ∇ ×⃗𝐅 =0 (this follows from Stokes's theorem). The converse is true on any Simply connected space.
• Fundamental theorem for line integrals may be used

Partially conservative field

As a consequence of Stokes's theorem, if a simply connected region is irrotational w.r.t. a field (i.e. ∀⃗𝐯 ∈𝑅 curl⁡⃗𝐅(⃗𝐯) =⃗𝟎), then the vector space is conservative within that region. However any region including a point which is not irrotational is not conservative. In other words, a closed path integral is 0 for any path whose enclosed region is irrotational everywhere.²

Practice problems

Practice problems are mostly for deriving a potential.

• 2023. Advanced Mathematical Methods, p. 28 (§1 problems 12–15)
• 2016. Calculus, pp. 1124–1135 (§16.3 exercises 3–19)
• 2016. Calculus, pp. 1149 (§16.5 exercises 13–18)

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