This is probably an obvious parallel that most people are aware of, but I only just noticed it the other day and it made me quite excited. The determinant in algebra has a lot in common with the integral in analysis. For example:
- They are both applied to functions, the integral to integrable functions, the determinant to linear transformations $T:V \rightarrow V$.
- They are both "sums of products."
- They both can be used to give a scalar result. (Not always, of course, but this is how they are first developed.)
- They are both important major structures in algebra and analysis.
- They are both defined in ways that feel 'backwards'-- the formal definition isn't always useful for calculating them-- then they come to represent multiple important concepts acting as a fulcrum in their fields. (ie. AREA is connected to ANTI-DERIVATIVES... or that SOLUTIONS TO AX=B are connected to LINEAR TRANSFORMATIONS.)
- They can both be used to give area and volume. (under a curve, or of a parallelepiped)
Question: What mathematical structure encompasses both? (If the answer is category theory, please go slowly with me, I don't understand that stuff yet.)
What else could we add to this list? Are there any problems or proofs that bring these parallels in to the light?
Are there any other mathematical structures that follow the pattern established by these two structures?
Were they developed independently (what I suspect) or is the determinant in some way patterned after the integral or vice versa? (I know my math history and have not come across anything about this.)