Fundamental Theorem of Line Integrals

Definition

The Fundamental Theorem of Line Integrals (FTLI) provides a method for evaluating the line integral of a conservative vector field over a path by using only the values of its potential function at the path’s endpoints. For a conservative field $\mathbf{F} = \nabla f$ and a smooth path $C$ from point $A$ to point $B$, the theorem states: $\int_C \nabla f \cdot d\mathbf{r} = f(B) - f(A)$

• How to read: “The line integral over the path C of the dot product of the gradient of f and d r equals f of B minus f of A.”
• Meaning: For a conservative field, work along any path from $A$ to $B$ depends only on the potential difference at endpoints — path geometry does not matter.

Why It Matters

This theorem is the rigorous justification for the concept of ‘potential energy’; it proves that in a conservative field (like gravity), your final state depends only on where you start and end, not the path you took, allowing us to simplify the physics of the entire universe into a series of ‘potential’ differences.

Core Concepts

• Potential Function: The scalar function $f$ whose gradient is the vector field $\mathbf{F}$.
• Endpoint Dependence: The integral’s value depends strictly on where the path starts ($A$) and ends ($B$).
• Conservative Fields: The theorem only applies to fields that are the gradient of some scalar function.
• Independence of Path: Because the result only involves $f(A)$ and $f(B)$, the specific geometry of the path $C$ between them does not affect the outcome.

Connected Concepts

• Fundamental Theorem of calculus, Part 1
• Fundamental Theorem of calculus, Part 2
• Comparison Properties of Integrals