Gradient theorem

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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.

If Template:Math is a differentiable function and Template:Mvar a differentiable curve in Template:Math which starts at a point Template:Math and ends at a point Template:Math, then

$$\underset{\gamma }{\int }\mathrm{\nabla }\phi (𝐫)\cdot \mathrm{d}𝐫=\phi \left(𝐪\right)-\phi \left(𝐩\right)$$

where Template:Math denotes the gradient vector field of Template:Math.

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 Template:Mvar as potential, Template:Math 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 Template:Mvar is a differentiable function from some open subset Template:Math to Template:Math and Template:Math is a differentiable function from some closed interval Template:Math to Template:Mvar (Note that Template:Math is differentiable at the interval endpoints Template:Math and Template:Math. To do this, Template:Math is defined on an interval that is larger than and includes Template:Math.), then by the multivariate chain rule, the composite function Template:Math is differentiable on Template:Math:

$$\frac{\mathrm{d}}{\mathrm{d}t}(\phi \circ 𝐫)(t)=\mathrm{\nabla }\phi (𝐫(t))\cdot {𝐫}^{\prime }(t)$$

for all Template:Mvar in Template:Math. Here the Template:Math denotes the usual inner product.

Now suppose the domain Template:Mvar of Template:Mvar contains the differentiable curve Template:Mvar with endpoints Template:Math and Template:Math. (This is oriented in the direction from Template:Math to Template:Math). If Template:Math parametrizes Template:Mvar for Template:Mvar in Template:Math (i.e., Template:Math represents Template:Mvar as a function of Template:Mvar), then

$$\begin{array}{cc}\underset{\gamma }{\int }\mathrm{\nabla }\phi (𝐫)\cdot \mathrm{d}𝐫& =\underset{a}{\overset{b}{\int }}\mathrm{\nabla }\phi (𝐫(t))\cdot {𝐫}^{\prime }(t)\mathrm{d}t\\ & =\underset{a}{\overset{b}{\int }}\frac{d}{dt}\phi (𝐫(t))\mathrm{d}t=\phi (𝐫(b))-\phi (𝐫(a))=\phi \left(𝐪\right)-\phi \left(𝐩\right),\end{array}$$

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 Template:Math is the circular arc oriented counterclockwise from Template:Math to Template:Math. Using the definition of a line integral,

$$\begin{array}{cc}\underset{\gamma }{\int }y\mathrm{d}x+x\mathrm{d}y& =\underset{0}{\overset{\pi -{\tan }^{-1}\left(\frac{3}{4}\right)}{\int }}((5\sin t)(-5\sin t)+(5\cos t)(5\cos t))\mathrm{d}t\\ & =\underset{0}{\overset{\pi -{\tan }^{-1}\left(\frac{3}{4}\right)}{\int }}25(-{\sin }^{2}t+{\cos }^{2}t)\mathrm{d}t\\ & =\underset{0}{\overset{\pi -{\tan }^{-1}\left(\frac{3}{4}\right)}{\int }}25\cos (2t)\mathrm{d}t={\frac{25}{2}\sin (2t)|}_0^{\pi -{\tan }^{-1}\left(\frac{3}{4}\right)}\\ & =\frac{25}{2}\sin (2\pi -2{\tan }^{-1}\left(\frac{3}{4}\right))\\ & =-\frac{25}{2}\sin \left(2{\tan }^{-1}\left(\frac{3}{4}\right)\right)=-\frac{25(3/4)}{(3/4{)}^2+1}=-12.\end{array}$$

This result can be obtained much more simply by noticing that the function $f(x,y)=xy$ has gradient $\mathrm{\nabla }f(x,y)=(y,x)$, so by the Gradient Theorem:

$$\underset{\gamma }{\int }y\mathrm{d}x+x\mathrm{d}y=\underset{\gamma }{\int }\mathrm{\nabla }(xy)\cdot (\mathrm{d}x,\mathrm{d}y)=xy{|}_{(5,0)}^{(-4,3)}=-4\cdot 3-5\cdot 0=-12.$$

Example 2

For a more abstract example, suppose Template:Math has endpoints Template:Math, Template:Math, with orientation from Template:Math to Template:Math. For Template:Math in Template:Math, let Template:Math denote the Euclidean norm of Template:Math. If Template:Math is a real number, then

$$\begin{array}{cc}\underset{\gamma }{\int }|𝐱{|}^{\alpha -1}𝐱\cdot \mathrm{d}𝐱& =\frac{1}{\alpha +1}\underset{\gamma }{\int }(\alpha +1)|𝐱{|}^{(\alpha +1)-2}𝐱\cdot \mathrm{d}𝐱\\ & =\frac{1}{\alpha +1}\underset{\gamma }{\int }\mathrm{\nabla }|𝐱{|}^{\alpha +1}\cdot \mathrm{d}𝐱=\frac{|𝐪{|}^{\alpha +1}-|𝐩{|}^{\alpha +1}}{\alpha +1}\end{array}$$

Here the final equality follows by the gradient theorem, since the function Template:Math is differentiable on Template:Math if Template:Math.

If Template:Math 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 Template:Math will fail to be defined there. However, the case Template:Math is somewhat different; in this case, the integrand becomes Template:Math, so that the final equality becomes Template:Math.

Note that if Template:Math, then this example is simply a slight variant of the familiar power rule from single-variable calculus.

Example 3

Suppose there are Template:Mvar point charges arranged in three-dimensional space, and the Template:Mvar-th point charge has charge Template:Math and is located at position Template:Math in Template:Math. We would like to calculate the work done on a particle of charge Template:Mvar as it travels from a point Template:Math to a point Template:Math in Template:Math. Using Coulomb's law, we can easily determine that the force on the particle at position Template:Math will be

$$𝐅(𝐫)=kq\sum _{i=1}^n\frac{Q_i(𝐫-{𝐩}_i)}{{|𝐫-{𝐩}_i|}^3}$$

Here Template:Math denotes the Euclidean norm of the vector Template:Math in Template:Math, and Template:Math, where Template:Math is the vacuum permittivity.

Let Template:Math be an arbitrary differentiable curve from Template:Math to Template:Math. Then the work done on the particle is

$$W=\underset{\gamma }{\int }𝐅(𝐫)\cdot \mathrm{d}𝐫=\underset{\gamma }{\int }\left(kq\sum _{i=1}^n\frac{Q_i(𝐫-{𝐩}_i)}{{|𝐫-{𝐩}_i|}^3}\right)\cdot \mathrm{d}𝐫=kq\sum _{i=1}^n(Q_i\underset{\gamma }{\int }\frac{𝐫-{𝐩}_i}{{|𝐫-{𝐩}_i|}^3}\cdot \mathrm{d}𝐫)$$

Now for each Template:Mvar, direct computation shows that

$$\frac{𝐫-{𝐩}_i}{{|𝐫-{𝐩}_i|}^3}=-\mathrm{\nabla }\frac{1}{|𝐫-{𝐩}_i|}.$$

Thus, continuing from above and using the gradient theorem,

$$W=-kq\sum _{i=1}^n(Q_i\underset{\gamma }{\int }\mathrm{\nabla }\frac{1}{|𝐫-{𝐩}_i|}\cdot \mathrm{d}𝐫)=kq\sum _{i=1}^n{Q}_i(\frac{1}{|𝐚-{𝐩}_i|}-\frac{1}{|𝐛-{𝐩}_i|})$$

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 Template:Math). 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 Template:Math is the gradient of some scalar-valued function (i.e., if Template:Math is conservative), then Template:Math is a path-independent vector field (i.e., the integral of Template:Math over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse: Template:Math theorem 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 Template:Math over every closed loop in the domain of Template:Math is zero, then Template:Math is the gradient of some scalar-valued function.

Proof of the converse

Suppose Template:Mvar is an open, path-connected subset of Template:Math, and Template:Math is a continuous and path-independent vector field. Fix some element Template:Math of Template:Mvar, and define Template:Math by$$f(𝐱):=\underset{\gamma [𝐚,𝐱]}{\int }𝐅(𝐮)\cdot \mathrm{d}𝐮$$Here Template:Math is any (differentiable) curve in Template:Mvar originating at Template:Math and terminating at Template:Math. We know that Template:Math is well-defined because Template:Math is path-independent.

Let Template:Math be any nonzero vector in Template:Math. By the definition of the directional derivative,$$\begin{array}{cc}\frac{\partial f(𝐱)}{\partial 𝐯}& =\underset{t\to 0}{\lim }\frac{f(𝐱+t𝐯)-f(𝐱)}{t}\\ & =\underset{t\to 0}{\lim }\frac{\underset{\gamma [𝐚,𝐱+t𝐯]}{\int }𝐅(𝐮)\cdot \mathrm{d}𝐮-\underset{\gamma [𝐚,𝐱]}{\int }𝐅(𝐮)\cdot d𝐮}{t}\\ & =\underset{t\to 0}{\lim }\frac{1}{t}\underset{\gamma [𝐱,𝐱+t𝐯]}{\int }𝐅(𝐮)\cdot \mathrm{d}𝐮\end{array}$$To calculate the integral within the final limit, we must parametrize Template:Math. Since Template:Math is path-independent, Template:Mvar is open, and Template:Mvar is approaching zero, we may assume that this path is a straight line, and parametrize it as Template:Math for Template:Math. Now, since Template:Math, the limit becomes$$\underset{t\to 0}{\lim }\frac{1}{t}\underset{0}{\overset{t}{\int }}𝐅(𝐮(s))\cdot {𝐮}^{\prime }(s)\mathrm{d}s=\frac{\mathrm{d}}{\mathrm{d}t}\underset{0}{\overset{t}{\int }}𝐅(𝐱+s𝐯)\cdot 𝐯\mathrm{d}s{|}_{t=0}=𝐅(𝐱)\cdot 𝐯$$where the first equality is from the definition of the derivative with a fact that the integral is equal to 0 at Template:Mvar = 0, and the second equality is from the first fundamental theorem of calculus. Thus we have a formula for Template:Math, (one of ways to represent the directional derivative) where Template:Math is arbitrary; for $f(𝐱):=\underset{\gamma [𝐚,𝐱]}{\int }𝐅(𝐮)\cdot \mathrm{d}𝐮$ (see its full definition above), its directional derivative with respect to Template:Math is$$\frac{\partial f(𝐱)}{\partial 𝐯}={\partial }_{𝐯}f(𝐱)=D_{𝐯}f(𝐱)=𝐅(𝐱)\cdot 𝐯$$where the first two equalities just show different representations of the directional derivative. According to the definition of the gradient of a scalar function Template:Math, $\mathrm{\nabla }f(𝐱)=𝐅(𝐱)$, thus we have found a scalar-valued function Template:Mvar whose gradient is the path-independent vector field Template:Math (i.e., Template:Math is a conservative vector field.), as desired.

Example of the converse principle

Template:Main 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 Template:Math is conservative (here Template:Mvar is some open, path-connected subset of Template:Math that contains a charge distribution). Following the ideas of the above proof, we can set some reference point Template:Math in Template:Mvar, and define a function Template:Math by

$$U_e(𝐫):=-\underset{\gamma [𝐚,𝐫]}{\int }{𝐅}_e(𝐮)\cdot \mathrm{d}𝐮$$

Using the above proof, we know Template:Math is well-defined and differentiable, and Template:Math (from this formula we can use the gradient theorem to easily derive the well-known formula for calculating work done by conservative forces: Template:Math). This function Template:Math is often referred to as the electrostatic potential energy of the system of charges in Template:Mvar (with reference to the zero-of-potential Template:Math). In many cases, the domain Template:Mvar is assumed to be unbounded and the reference point Template:Math is taken to be "infinity", which can be made rigorous using limiting techniques. This function Template:Math 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

$$\underset{\partial \gamma }{\int }\varphi =\underset{\gamma }{\int }\mathrm{d}\varphi$$

for any 0-form, Template:Mvar, defined on some differentiable curve Template:Math (here the integral of Template:Math over the boundary of the Template:Mvar is understood to be the evaluation of Template:Math 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 Template:Mvar over the boundary of some orientable manifold Template:Math is equal to the integral of its exterior derivative Template:Math over the whole of Template:Math, i.e.,

$$\underset{\partial \mathrm{\Omega }}{\int }\omega =\underset{\mathrm{\Omega }}{\int }\mathrm{d}\omega$$

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 Template:Mvar is a form defined on a contractible domain, and the integral of Template:Mvar over any closed manifold is zero. Then there exists a form Template:Mvar such that Template:Math. 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
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