Pronunciation: [pʊʃfˈɔːwəd] (IPA) 
The word "pushforward" is not a commonly used term, but in mathematics, it refers to a function that takes a point from one space to another. Its spelling can be explained using the International Phonetic Alphabet (IPA) as /’pʊʃfɔːwəd/. It starts with the sound /p/ followed by /ʊ/ as in "put," then /ʃ/ as in "shoe," and then the sound /f/ followed by /ɔː/ as in "fork." Lastly, it ends with the sound /w/ and /əd/ as in "cupboard."
Pushforward is a term used in mathematics and physics to describe the operation of mapping a point or vector from one space to another using a given function. It is a concept commonly used in the study of differential geometry, manifold theory, and algebraic geometry.
More specifically, in differential geometry, the pushforward is a mapping between tangent spaces induced by a differentiable function. Given a smooth function between two manifolds, the pushforward takes a tangent vector from the first manifold and maps it to a tangent vector on the second manifold. This is done by the derivative of the function at the corresponding point. In other words, the pushforward is a linear approximation of how the function changes nearby tangent vectors.
In algebraic geometry, the pushforward is used to extend the concept of mapping between spaces defined by polynomials or algebraic equations. It allows for the study of how divisors, cohomology, and other mathematical structures change when a morphism is applied.
Overall, the pushforward is a powerful tool for understanding the transformation of points, vectors, structures, and properties under given functions or mappings. It provides a way to analyze the propagation of information or attributes from one mathematical object to another.
The word "pushforward" comes from the combination of two words: "push" and "forward".
The term "push" originated from the Old English word "puscan", meaning to exert force or pressure in order to move something. It has remained relatively unchanged in meaning and pronunciation over time.
The word "forward" has roots in the Old English word "foreweard", which combines "fore", meaning in front or ahead, with "weard", meaning toward or facing. It implies movement in a directed or progressing manner towards a specific location or objective.
When these two words are combined, "pushforward" denotes the act of applying force or pressure to drive something ahead or in a forward direction. It is commonly used in various contexts, including physics, mathematics, and general language.
