Gradient theorem

The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a <a href="/facts/Line_integral/10s4B3Nn">line integral</a> through a <a href="/facts/Conservative_vector_field/MoS3V9aV">gradient field</a> can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the <a href="/facts/Fundamental_theorem_of_calculus/pHeSGtUx">second fundamental theorem of calculus</a> to any curve in a plane or space (generally n-dimensional) rather than just the real line.
If φ : U ⊆ Rn → R is a <a href="/facts/Differentiable_function/jQqTgLk1">differentiable function</a> and γ a <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> 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
 
 )
 
 
 
 {\displaystyle \int _{\gamma }\nabla \varphi (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} =\varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)}

where ∇φ denotes the gradient vector field of φ.
The gradient theorem implies that line integrals through gradient fields are <a href="/facts/Conservative_vector_field/MoS3V9aV">path-independent</a>. In physics this theorem is one of the ways of defining a <a href="/facts/Conservative_force/xY5QmC4q">conservative force</a>. By placing φ as potential, ∇φ is a <a href="/facts/Conservative_field/MoS3V9aV">conservative field</a>. <a href="/facts/Work_(physics)/mbclFPBs">Work</a> 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 <a href="/facts/Scalar_field/8MtPTGob">scalar field</a>. Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics.

Gradient theorem open-in-new

Gradient theorem