structural:expansion, quality:correction, structural:reformatting
Path independence

If a vector field F is the gradient of a scalar field G (i.e. if F is conservative), that is, F = ∇ G, {\displaystyle \mathbf {F} =\nabla G,} then by the multivariable chain rule the derivative of the composition of G and r (t) is d G (r (t)) d t = ∇ G (r (t)) ⋅ r ′ (t) = F (r (t)) ⋅ r ′ (t) {\displaystyle {\frac {dG(\mathbf {r} (t))}{dt}}=\nabla G(\mathbf {r} (t))\cdot \mathbf {r} '(t)=\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)} which happens to be the integrand for the line integral of F on r (t). It follows, given a path C, that ∫ C F (r) ⋅ d r = ∫ a b F (r (t)) ⋅ r ′ (t) d t = ∫ a b d G (r (t)) d t d t = G (r (b)) − G (r (a)). {\displaystyle \int _{C}\mathbf {F} (\mathbf {r})\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt=\int _{a}^{b}{\frac {dG(\mathbf {r} (t))}{dt}}\,dt=G(\mathbf {r} (b))-G(\mathbf {r} (a)).}

In other words, the integral of F over C depends solely on the values of G at the points r (b) and r (a), and is thus independent of the path between them. For this reason, a line integral of a conservative vector field is called path independent.