Gradient theorem

Gradient theorem
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. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line.
For as a differentiable function and as any continuous curve in which starts at a point and ends at a point , then
where denotes the gradient vector field of .
The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a conservative force. By placing as potential, is a conservative field. 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.
The gradient theorem also has an interesting converse: any path-independent vector field can be expressed as the gradient of a scalar field. Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics.
Proof
If is a differentiable function from some open subset to and is a differentiable function from some closed interval to (Note that is differentiable at the interval endpoints and . To do this, is defined on an interval that is larger than and includes .), then by the multivariate chain rule, the composite function is differentiable on :
for all in . Here the denotes the usual inner product.
Now suppose the domain of contains the differentiable curve with endpoints and . (This is oriented in the direction from to ). If parametrizes for in (i.e., represents as a function of ), then
where the definition of a line integral is used in the first equality, the above equation is used in the second equality, and the second fundamental theorem of calculus is used in the third equality.
Even if the gradient theorem (also called fundamental theorem of calculus for line integrals) has been proved for a differentiable (so looked as smooth) curve so far, the theorem is also proved for a piecewise-smooth curve since this curve is made by joining multiple differentiable curves so the proof for this curve is made by the proof per differentiable curve component.
Examples
Example 1
Suppose is the circular arc oriented counterclockwise from to . Using the definition of a line integral,
This result can be obtained much more simply by noticing that the function has gradient , so by the Gradient Theorem:
Example 2
For a more abstract example, suppose has endpoints with orientation from to . For in , let denote the Euclidean norm of . If is a real number, then
Here the final equality follows by the gradient theorem, since the function is differentiable on if .
If 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 will fail to be defined there. However, the case is somewhat different; in this case, the integrand becomes , so that the final equality becomes .
Note that if , then this example is simply a slight variant of the familiar power rule from single-variable calculus.
Example 3
Suppose there are point charges arranged in three-dimensional space, and the -th point charge has charge and is located at position in . We would like to calculate the work done on a particle of charge as it travels from a point to a point in . Using Coulomb's law, we can easily determine that the force on the particle at position will be
Here denotes the Euclidean norm of the vector in , and , where is the vacuum permittivity.
Let be an arbitrary differentiable curve from to . Then the work done on the particle is
Now for each , direct computation shows that
Thus, continuing from above and using the gradient theorem,
We are finished. Of course, we could have easily completed this calculation using the powerful language of electrostatic potential or electrostatic potential energy (with the familiar formulas ). However, we have not yet defined potential or potential energy, because the converse of the gradient theorem is required to prove that these are well-defined, differentiable functions and that these formulas hold (see below). Thus, we have solved this problem using only Coulomb's Law, the definition of work, and the gradient theorem.
Converse of the gradient theorem
The gradient theorem states that if the vector field is the gradient of some scalar-valued function (i.e., if is conservative), then is a path-independent vector field (i.e., the integral of over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:
It is straightforward to show that a vector field is path-independent if and only if the integral of the vector field over every closed loop in its domain is zero. Thus the converse can alternatively be stated as follows: If the integral of over every closed loop in the domain of is zero, then is the gradient of some scalar-valued function.
Proof of the converse
Suppose is an open, path-connected subset of , and is a continuous and path-independent vector field. Fix some element of , and define byHere is any (differentiable) curve in originating at and terminating at . We know that is well-defined because is path-independent.
Let be any nonzero vector in . By the definition of the directional derivative,To calculate the integral within the final limit, we must parametrize . Since is path-independent, is open, and is approaching zero, we may assume that this path is a straight line, and parametrize it as for . Now, since , the limit becomeswhere the first equality is from the definition of the derivative with a fact that the integral is equal to 0 at = 0, and the second equality is from the first fundamental theorem of calculus. Thus we have a formula for , (one of ways to represent the directional derivative) where is arbitrary; for (see its full definition above), its directional derivative with respect to iswhere the first two equalities just show different representations of the directional derivative. According to the definition of the gradient of a scalar function thus we have found a scalar-valued function whose gradient is the path-independent vector field (i.e., is a conservative vector field.), as desired.
Example of the converse principle
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 is conservative (here is some open, path-connected subset of that contains a charge distribution). Following the ideas of the above proof, we can set some reference point in , and define a function by
Using the above proof, we know is well-defined and differentiable, and (from this formula we can use the gradient theorem to easily derive the well-known formula for calculating work done by conservative forces: ). This function is often referred to as the electrostatic potential energy of the system of charges in (with reference to the zero-of-potential ). In many cases, the domain is assumed to be unbounded and the reference point is taken to be "infinity", which can be made rigorous using limiting techniques. This function is an indispensable tool used in the analysis of many physical systems.
Generalizations
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
for any 0-form defined on some differentiable curve (here the integral of over the boundary of the is understood to be the evaluation of at the endpoints of γ).
Notice the striking similarity between this statement and the generalized Stokes’ theorem, which says that the integral of any compactly supported differential form over the boundary of some orientable manifold is equal to the integral of its exterior derivative over the whole of , i.e.,
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. In particular, suppose is a form defined on a contractible domain, and the integral of over any closed manifold is zero. Then there exists a form such that . Thus, on a contractible domain, every closed form is exact. This result is summarized by the Poincaré lemma.
See also
State function
Scalar potential
Jordan curve theorem
Differential of a function
Classical mechanics
References
Theorems in calculus
Articles containing proofs