What is dx?

Calculus teacher: It reminds you that x is the variable in the integration.

Private math. tutor: Points will be deducted if one forgets to write it down.

Engineer: It is an infinitesimally small \Delta x.

Teacher of advanced calculus: It is the differential of the function x.

Measure theorist: We usually write d \mu instead.

Differential geometer: It is a smooth function mapping a point P in a manifold to an element in the cotangent space at P.

Algebraic geometer: Let A be a k-algebra over a field k. dx is the equivalent class 1\otimes_k x + S in the quotient (A\otimes_k A )/ S, where S is the A-submodule of A\otimes_k A generated by elements of the form x \otimes_k yz - xy\otimes_k z - xz \otimes_k y. In general, f \,dx is f \otimes_k x + S.

 

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