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
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.
The gradienttheorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated...
the endpoints of the path, and can be evaluated by the gradienttheorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous)...
and a terminal point B {\displaystyle B} . Then the gradienttheorem (also called fundamental theorem of calculus for line integrals) states that ∫ P v...
areas, and calculation of gradients) are actually closely related. From the conjecture and the proof of the fundamental theorem of calculus, calculus as...
div generalize immediately to other dimensions, as do the gradienttheorem, divergence theorem, and Laplacian (yielding harmonic analysis), while curl and...
algorithm for finding the minimum of a function Gradienttheorem, theorem that a line integral through a gradient field can be evaluated by evaluating the original...
Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e...
(\mathbf {p} )=\int _{P}\nabla \psi \cdot d{\boldsymbol {\ell }}} (gradienttheorem) A | ∂ P = A ( q ) − A ( p ) = ∫ P ( d ℓ ⋅ ∇ ) A {\displaystyle \mathbf...
_{C}\nabla U'\cdot d\mathbf {x} ,} which can be evaluated using the gradienttheorem to obtain W = U ′ ( x B ) − U ′ ( x A ) . {\displaystyle W=U'(\mathbf...
embodied by the integral theorems of vector calculus:: 543ff Gradienttheorem Stokes' theorem Divergence theorem Green's theorem. In a more advanced study...
correlation, in order to understand the origin of specific electric field gradients in crystals. Despite recent improvements, there are still difficulties...
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for finding a local minimum of a differentiable...
conditions represents the fundamental theorem of the gradient and is true for any vector field that is a gradient of a differentiable single valued scalar...
making V E {\textstyle V_{\mathbf {E} }} well-defined everywhere. The gradienttheorem then allows us to write: E = − ∇ V E {\displaystyle \mathbf {E} =-\mathbf...
mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood...
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through...
generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about...
= − ∇ → ϕ . {\displaystyle {\vec {E}}=-{\vec {\nabla }}\phi .} The gradienttheorem can be used to establish that the electrostatic potential is the amount...
\mathbf {u} } is path-independent. Finally, by the converse of the gradienttheorem, a scalar function ψ ( x , y , t ) {\displaystyle \psi (x,y,t)} exists...
with respect to x {\displaystyle x} and y {\displaystyle y} . The gradienttheorem asserts that a 1-form is exact if and only if the line integral of...
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