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 𝐶 be a smooth curve with endpoints ⃗𝐚 and ⃗𝐛. Let 𝑓 be a differentiable function whose gradient grad⁡𝑓 is continuous on 𝐶. Then¹ calculus

∫𝐶⃗∇𝑓⋅𝑑⃗𝐫=𝑓(⃗𝐚)−𝑓(⃗𝐛)

An important consequence of this is that if 𝐶 is closed, then the integral evaluates to 0. This is the defining property of a Conservative vector field, defined as grad⁡𝑓.

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Footnotes

1. 2016. Calculus, p. 1127 ↩