How Does a Line Integral of a Scalar Field Differ from a Regular Integral?

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SUMMARY

The discussion clarifies the distinction between line integrals of scalar fields and regular integrals. A line integral computes the integral of a scalar field along a specific path, while a regular integral evaluates the total accumulation over a defined interval. In one-dimensional cases, both integrals yield the same result, but in higher dimensions, the choice of path significantly influences the outcome of the line integral due to the existence of multiple paths between two points.

PREREQUISITES
  • Understanding of scalar fields
  • Familiarity with line integrals
  • Knowledge of regular integrals
  • Basic concepts of multidimensional calculus
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Students and professionals in mathematics, physics, and engineering who seek to deepen their understanding of calculus concepts, particularly in the context of scalar and vector fields.

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Okay this might be a nooby question, but it bothers me.

What is the difference between the line integral of a scalar field and just a regular integral over the scalar field?

For a function of one variable i certainly can't see the difference. But then I thought they might be identical in this case. Is that true?
Because then I can see that in n dimensions where n≠1, you can generally choose infinately many paths between 2 points.
 
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If the "field" is one dimensional, there cannot be a difference. If the field has dimension 2 or higher, there are an infinite number of different paths through the field.
 

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