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Question: Why is the potential function defined differently in physics and calculus?
Answer: After reading further on in my calculus textbook, I found that they later defined the potential energy as the negative of the potential function.

So it appears that the reasoning presented in my question was correct, but that some sources simply use a slightly different definition for "potential function" – not because it corresponds to potential energy, but because a convenient name is needed for it.

Wikipedia says that the gradient of the potential is the negative of the vector field, but that "In some cases, mathematicians may use a positive sign in front of the gradient to define the potential."