# Using Stokes' Theorem with a Constant Scalar Field The problem illustrates the significance of Generalized Stokes' Theorem in relating surface integrals to line integrals, particularly emphasizing that the scalar field $$\phi(x)$$ must be constant on the boundary curve $$C$$. This constancy allows for simplification of the line integral, transforming it into a fundamental case where the Fundamental Theorem of Line Integrals applies, ensuring that the integral evaluates to zero. The visual demonstration reinforces that without the boundary condition of a constant scalar field, the integral can yield non-zero results, underscoring the critical role of this condition in achieving a predictable outcome. ### :clapper:A constant scalar field leads to a zero integral result > The demo visually confirms that the condition of a constant scalar field on the boundary is essential for the surface integral to be zero. When the scalar field $$\phi$$ is constant on the boundary, the line integral evaluates to zero. When $$\phi$$ is not constant, the line integral has a non-zero value, and the proof fails. This highlights the importance of the initial condition in the problem statement, which turns a potentially complex integral into a straightforward case with a predictable result. {% embed url="" %} ### :pen\_ballpoint:Mathematical Proof {% embed url="" %}