Fundamental theorem of calculus

Fundamental theorem for line integrals

The fundamental theorem for line integrals is a generalisation of the Fundamental theorem of calculus describing vector fields defined as potentials. It essentially states that the line integral and multivariable gradients are inverse operations.

Let be a smooth curve with endpoints and . Let be a differentiable function whose gradient is continuous on . Then¹ calculus

An important consequence of this is that if is closed, then the integral evaluates to . This is the defining property of a Conservative vector field, defined as .

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Footnotes

1. 2016. Calculus, p. 1127 ↩