Question

State the Fundamental Theorem of Line Integrals.

Short answer

The Fundamental Theorem of Line Integrals states that if a particle moves in a vector field \( \mathbf{F} = \nabla f \) along a curve \( C \) from point \( A \) to point \( B \), the work done by the field on the particle is equal to the change in potential: \( \int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A) \).

Step-by-step solution

Understanding the Theorem

The Fundamental Theorem of Line Integrals states that if a particle moves in a vector field \( \mathbf{F} = \nabla f \) (where \( \nabla f \) is a gradient of a scalar function \( f \)) along a curve \( C \) from point \( A \) to point \( B \), the work done by the field on the particle is equal to the change in potential: \( \int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A) \). This equation shows the integral of the field \(\mathbf{F}\) over the curve \(C\) is equal to the difference in the values of the function \( f \) at points \( B \) and \( A \).