The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
In physics this theorem is one of the ways of defining a conservative force.
Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows.
Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics.
), then by the multivariate chain rule, the composite function φ ∘ r is differentiable on [a, b]:
Now suppose the domain U of φ contains the differentiable curve γ with endpoints p and q.
[2] Suppose γ ⊂ R2 is the circular arc oriented counterclockwise from (5, 0) to (−4, 3).
For a more abstract example, suppose γ ⊂ Rn has endpoints p, q, with orientation from p to q.
Here the final equality follows by the gradient theorem, since the function f(x) = |x|α+1 is differentiable on Rn if α ≥ 1.
If α < 1 then this equality will still hold in most cases, but caution must be taken if γ passes through or encloses the origin, because the integrand vector field |x|α − 1x will fail to be defined there.
Note that if n = 1, then this example is simply a slight variant of the familiar power rule from single-variable calculus.
Using Coulomb's law, we can easily determine that the force on the particle at position r will be
Here |u| denotes the Euclidean norm of the vector u in R3, and k = 1/(4πε0), where ε0 is the vacuum permittivity.
Thus, we have solved this problem using only Coulomb's Law, the definition of work, and the gradient theorem.
The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points).
Thus the converse can alternatively be stated as follows: If the integral of F over every closed loop in the domain of F is zero, then F is the gradient of some scalar-valued function.
To calculate the integral within the final limit, we must parametrize γ[x, x + tv].
Since F is path-independent, U is open, and t is approaching zero, we may assume that this path is a straight line, and parametrize it as u(s) = x + sv for 0 < s < t. Now, since u'(s) = v, the limit becomes
Thus we have a formula for ∂vf, (one of ways to represent the directional derivative) where v is arbitrary; for
[3] To illustrate the power of this converse principle, we cite an example that has significant physical consequences.
In classical electromagnetism, the electric force is a path-independent force; i.e. the work done on a particle that has returned to its original position within an electric field is zero (assuming that no changing magnetic fields are present).
Therefore, the above theorem implies that the electric force field Fe : S → R3 is conservative (here S is some open, path-connected subset of R3 that contains a charge distribution).
Following the ideas of the above proof, we can set some reference point a in S, and define a function Ue: S → R by
Using the above proof, we know Ue is well-defined and differentiable, and Fe = −∇Ue (from this formula we can use the gradient theorem to easily derive the well-known formula for calculating work done by conservative forces: W = −ΔU).
In many cases, the domain S is assumed to be unbounded and the reference point a is taken to be "infinity", which can be made rigorous using limiting techniques.
This function Ue is an indispensable tool used in the analysis of many physical systems.
Many of the critical theorems of vector calculus generalize elegantly to statements about the integration of differential forms on manifolds.
In the language of differential forms and exterior derivatives, the gradient theorem states that
This powerful statement is a generalization of the gradient theorem from 1-forms defined on one-dimensional manifolds to differential forms defined on manifolds of arbitrary dimension.
The converse statement of the gradient theorem also has a powerful generalization in terms of differential forms on manifolds.