Why is this legitimate notation?

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Question about Rainman’s sum and continuity
r/mathematics is a subreddit dedicated to focused questions and discussion concerning mathematics.When may we consider a variable force as constant after using limit of Riemann sums (where across each slice, after the limiting process, the force is constant), and when can we not do this? Thanks!
Question about when we can treat force which is variable, as constant in integration based on this snapshot

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When we push a mower, why is the force shown downward-forward at a 45? To me, intuition says we are pushing perfectly parallel to the ground as we hold the handles, but every physics diagram I see shows a downward-forward 45 degrees along the angle
Anyone have any idea why the third diagram would be devoid of current in the common area? It’s really baffling me.
A subreddit for practical questions about component-level electronic circuits: design, repair, component buying, test gear and tools.How to prove the “trivial?” portion not addressed by this proof author?

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On standard analysis and physicists
Welcome to r/calculus - a space for learning calculus and related disciplines. Remember to read the rules before posting and flair your posts appropriately.On standard analysis and physicists
r/mathematics is a subreddit dedicated to focused questions and discussion concerning mathematics.Physicists and their love of differentials

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5 snapshots provided; need help with derivation
This subreddit is for questions of a mathematical nature. Please read the subreddit rules below before posting.Proof question
r/mathematics is a subreddit dedicated to focused questions and discussion concerning mathematics.How is equating (dv/dt)dx with (dx/dt)dv justified in these pics
This subreddit is for questions of a mathematical nature. Please read the subreddit rules below before posting.[physics 2] work for dipole in uniform electric field
A place for physics students of any level to discuss the intricate profoundness of the universe.HV lines can produce voltage well above what is needed to run an incandescent light bulb, so why won’t it glow under a HV line but a fluorescent will?
Welcome to /r/AskTechnology! Ask Questions. Get Answers. Discuss Technology. Please keep discussions civil, on topic, free of anecdotes, free of speculation, advertisement and marketing-free.[Physics 2] why can this parallel portion be considered negligible/nonexistent

A place for physics students of any level to discuss the intricate profoundness of the universe.