Suppose a particle travels a path γ:I⊂R→R3 subject to a force F:R3→TR3, then we know that we define the work done by the force as
W=∫γF=∫I⟨F∘γ,γ′⟩
Now usually I see the term "work done against a force" and I don't really understand what it means. The reason is that in my understanding, work is always done by a force upon a particle or system of particles. If we talk about work done against a force, it is work done by which force on which particle or system?
Also, mathematically, how it is obtained? If we want to know not the work done by a force, but against it how we obtain it? I imagine is just the opposite, just changing the sign, but I'm unsure, and if it is really that I can't grasp why it should be.
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