The line integral is constructed analogously to the Riemann integral and it exists if the curve is rectifiable (has finite length) and the vector field is continuous.
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The data may be scalar (such as the concentration of a chemical agent in the brain), vector or tensor fields (like the displacement or strain tensor fields when the gear is in action) at different points of the object.
Thirring–Wess model describing the interaction of a Dirac field with a vector field in dimension two