In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.
Read more about Zariski Tangent Space: Example: Plane Curve, Definition, Example Extended, Properties
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—Freya Stark (1893–1993)