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Diffeology is a branch of mathematics that generalizes the notion of smooth manifolds. It was introduced by Jean-Marie Dufour and his collaborators in the 1980s to provide a more flexible framework for studying smooth structures on spaces that may not have a well-defined manifold structure. In traditional differential geometry, a smooth manifold is defined as a topological space that locally resembles Euclidean space and has a compatible smooth structure.

A differentiable curve is a mathematical concept referring to a curve that can be described by a differentiable function. In a more formal sense, a curve is said to be differentiable if it is possible to compute its derivative at every point in its domain. For a curve defined in a two-dimensional space, represented by a function \( y = f(x) \), it is differentiable at a point if the derivative \( f'(x) \) exists at that point.

A **differentiable stack** is a concept arising from the fields of differential geometry, algebraic topology, and category theory, particularly in the context of homotopy theory and advanced mathematical frameworks like derived algebraic geometry. In general, a **stack** is a categorical structure that allows for the systematic handling of "parametrized" objects, facilitating the study of moduli problems in algebraic geometry and related fields.

Differential forms are an essential concept in differential geometry and mathematical analysis. They generalize the idea of functions and can be used to describe various physical and geometric phenomena, particularly in the context of calculus on manifolds. Here's an overview of what differential forms are: ### Definition: A **differential form** is a mathematical object that is fully defined on a differentiable manifold.

A **differential invariant** is a property or quantity in differential geometry that remains unchanged under particular types of transformations, usually involving differentiable functions or mappings. These invariants play a crucial role in studying geometric objects and their properties without being affected by the coordinate system or parameterization used to describe them.

As of my last update in October 2023, "Diffiety" does not appear to be a widely recognized term in academic or popular culture. It's possible that it could be a misspelling, a new concept, or a niche term that has emerged after my last update.

Dirac structure refers to a mathematical framework used in the context of quantum mechanics and quantum field theory, particularly within the realm of Dirac's formulation of quantum mechanics. It is associated with the treatment of spinor fields, which are essential for describing particles with spin, such as electrons.

Discrete differential geometry is a branch of mathematics that studies geometric structures and concepts using discrete analogs rather than continuous ones. It often focuses on the analysis and approximation of geometric properties of surfaces and spaces through polygonal or polyhedral representations, as opposed to smooth manifolds that are typically the focus of classical differential geometry.

In differential geometry, the term "distribution" refers to a smooth assignment of a subspace of the tangent space at each point of a manifold. More formally, given a smooth manifold \( M \), a distribution is a smooth assignment of a vector subspace \( D_p \) of the tangent space \( T_p M \) at each point \( p \in M \). Distributions are often used to study geometric structures, such as foliations and control systems.

The double tangent bundle is a mathematical construction in differential geometry that generalizes the notion of tangent bundles. To understand the double tangent bundle, we first need to comprehend what a tangent bundle is. ### Tangent Bundle For a smooth manifold \( M \), the tangent bundle \( TM \) is a vector bundle that consists of all tangent vectors at every point on the manifold.

A double vector bundle is a mathematical structure that arises in differential geometry and algebraic topology. It generalizes the concept of a vector bundle by considering not just one vector space associated with each point in a manifold, but two layers of vector spaces.

In mathematics, particularly in the fields of convex analysis and differential geometry, the term "dual curve" can refer to different concepts depending on the context in which it is used. Here are a few interpretations: 1. **Dual Curves in Projective Geometry**: In projective geometry, the duality principle states that points and lines can be interchanged. The dual curve of a given curve can be constructed where each point on the dual curve represents a line tangent to the original curve.

A Dupin hypersurface is a specific type of hypersurface in differential geometry characterized by certain properties of its principal curvatures. More formally, a hypersurface in a Riemannian manifold is called a Dupin hypersurface if its principal curvatures are constant along the principal curvature directions.

Dynamic fluid film equations are mathematical formulations that describe the behavior of thin films of fluid that flow under the influence of various forces, such as gravity, surface tension, and viscous forces. These equations are crucial in understanding phenomena in various fields, including materials science, engineering, and fluid dynamics. In general, a fluid film can be considered a thin layer of fluid with a small thickness compared to its lateral dimensions.

Eguchi-Hanson space is a specific example of a Ricci-flat manifold that arises in the study of gravitational theories in higher dimensions, particularly in the context of string theory and differential geometry. It is a four-dimensional, asymptotically locally Euclidean manifold that can be described as follows: 1. **Metric Structure**: The Eguchi-Hanson space can be understood as a self-dual solution to the Einstein equations with a negative cosmological constant.

An elliptic complex is a concept in the field of mathematics, specifically within the areas of partial differential equations and the theory of elliptic operators. It relates to elliptic differential operators and the mathematical structures associated with them. ### Key Concepts: 1. **Elliptic Operators**: These are a class of differential operators that satisfy a certain condition (the ellipticity condition), which ensures the well-posedness of boundary value problems. An operator is elliptic if its principal symbol is invertible.

In mathematics, the term "envelope" can refer to a variety of concepts depending on the context in which it is used. Here are a few common interpretations: 1. **Envelope of a Family of Curves**: The envelope of a family of curves is a curve that is tangent to each member of the family at some point.

An Equiareal map, also known as an equal-area map, is a type of map projection that maintains the consistency of area proportions across the entire map. This means that regions on the map are represented in the same area ratio as they are on the Earth's surface. As a result, if two areas are equal in size on the map, they will also be equal in size in reality, regardless of their location.

Equivalent latitude is a concept used in atmospheric science and meteorology to describe the latitude corresponding to a particular atmospheric condition or property that is typically associated with a certain latitude in the atmosphere. It is often used in the context of phenomena such as the stratosphere, tropopause, or specific atmospheric trace gases. One common application of equivalent latitude is in the study of the ozone layer and the polar vortex.

Equivariant differential forms are a specific type of differential forms that respect certain symmetries in a mathematical or physical context, particularly in the fields of differential geometry and algebraic topology. These forms are often associated with group actions on manifolds, where the structure of the manifold and the properties of the forms are invariant under the action of a group.