梯度定理

设函数 φ:U⊆ R ⁿ→ R，则 φ (q)-φ (p)=∫ _(γ [p, q ])∇ φ (r)· dr.

梯度定理把微积分基本定理从直线数轴推广到平面、空间，乃至一般的 n 维空间中的曲线。

梯度定理表明梯度场的曲线积分是路径无关的，这是物理学中“保守力”的定义方式之一。如果 φ 是位势，则 ∇ φ 就是保守向量场。上面的公式表明：保守力做功只和物体运动路径的端点有关，而与路径本身无关。

梯度定理有个逆定理，是说任何路径无关的向量场都可以表示为某个标量场的梯度。这个逆定理和原定理一样在纯粹和应用数学中有很多推论和应用。

证明

设 φ 是个从 R ⁿ 中的开集 U 到 R 的可微函数，设 r 是闭区间 [a,b] 到 U 的可微函数，那么由多元复合函数求导法则，复合函数 φ ∘ r 在闭区间 [a,b] 上可微，并且对所有 t∈ [a,b]， d/dt(φ ∘ r)(t)=∇ φ (r (t))· r '(t)

这里 · 是 R ⁿ 上的内积。 φ 的定义域 U 中含有从p到q的可微曲线γ，定向为从p至q。设 r (t) 是γ的参数化（其中 t∈ [a,b]），那么上面的式子说明 （公式略）

第一个等式是根据曲线积分的定义，第三个等式用了微积分基本定理。

梯度定理的逆定理

梯度定理说明如果一个向量场 F 是某个标量函数的梯度（即保守场），则 F 是路径无关的（即 F 沿分段可微的曲线的积分只和路径的端点有关）。这个定理有个强大的逆定理，是说若 F 是个路径无关的向量场，则它是某个标量函数的梯度。容易证明一个向量场是路径无关的当且仅当它沿任何闭曲线积分为零，因此梯度定理的逆定理是说如果 F 沿定义域中的任何闭曲线积分为零，则它是某标量函数的梯度。

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 φ: U ⊆ Rn → R is a differentiable function and γ a differentiable curve in U which starts at a point p and ends at a point q, then ∫ _(γ)∇ φ (r)· d r =φ (q)-φ (p)

where ∇φ denotes the gradient vector field of φ.

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 φ as potential, ∇φ 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 φ is a differentiable function from some open subset U ⊆ Rn to R and r is a differentiable function from some closed interval [a, b] to U (Note that r is differentiable at the interval endpoints a and b. To do this, r is defined on an interval that is larger than and includes [a, b].), then by the multivariate chain rule, the composite function φ ∘ r is differentiable on [a, b]: d/(d t)(φ ∘ r)(t)=∇ φ (r (t))· r '(t)

for all t in [a, b]. Here the ⋅ denotes the dot product.

Now suppose the domain U of φ contains the differentiable curve γ with endpoints p and q. (This is oriented in the direction from p to q). If r parametrizes γ for t in [a, b] (i.e., r represents γ as a function of t), then （公式略）

Examples

Example 1

Suppose γ ⊂ R2 is the circular arc oriented counterclockwise from (5, 0) to (−4, 3). Using the definition of a line integral, （公式略）

This result can be obtained much more simply by noticing that the function f(x,y)=xy has gradient ∇ f(x,y)=(y,x), so by the Gradient Theorem: （公式略）

Example 2

For a more abstract example, suppose γ ⊂ Rn has endpoints p, q, with orientation from p to q. For u in Rn, let |u| denote the Euclidean norm of u. If α ≥ 1 is a real number, then （公式略）

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.

Converse of 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). 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 F over every closed loop in the domain of F is zero, then F is the gradient of some scalar-valued function.

Proof of the converse

Suppose U is an open, path-connected subset of Rn, and F: U → Rn is a continuous and path-independent vector field.

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 ∫ _(∂ γ)φ =∫ _(γ)d φ

for any 0-form, ϕ, defined on some differentiable curve γ ⊂ Rn (here the integral of ϕ over the boundary of the γ is understood to be the evaluation of ϕ 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 ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e., ∫ _(∂ Ω)ω =∫ _(Ω)d ω