# Quick Calculation 3.10

A First Course in String Theory

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Equation (3.96) has content: if you move a particle along a closed loop in a static gravitational field, the net work that you do against the gravitational field is zero.

Prove the above statement.

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Eq. (3.96):

$\displaystyle{\vec g = - \nabla V_g}$

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By the gradient theorem (aka the fundamental theorem of calculus for line integrals):

$\displaystyle{\int _{\gamma }\nabla \varphi (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} =\varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)}$

— Me@2022-07-03 04:27:19 PM

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