Pseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.
Dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
Cross productIn mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.
Principal bundleIn mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with An action of on , analogous to for a product space. A projection onto . For a product space, this is just the projection onto the first factor, . Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of .
Triple productIn geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two.
Product lifecycleIn industry, product lifecycle management (PLM) is the process of managing the entire lifecycle of a product from its inception through the engineering, design and manufacture, as well as the service and disposal of manufactured products. PLM integrates people, data, processes, and business systems and provides a product information backbone for companies and their extended enterprises. The inspiration for the burgeoning business process now known as PLM came from American Motors Corporation (AMC).
Product managerA product manager (PM) is a professional role that is responsible for the development of products for an organization, known as the practice of product management. Product managers own the product strategy behind a product (physical or digital), specify its functional requirements, and manage feature releases. Product managers coordinate work done by many other functions (like software engineers, data scientists, and product designers), and are ultimately responsible for product outcomes.
Euler characteristicIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by (Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids.
Product managementProduct management is the business process of planning, developing, launching, and managing a product or service. It includes the entire lifecycle of a product, from ideation to development to go to market. Product managers are responsible for ensuring that a product meets the needs of its target market and contributes to the business strategy, while managing a product or products at all stages of the product lifecycle. Software product management adapts the fundamentals of product management for digital products.
Immersion (mathematics)In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f : M → N is an immersion if is an injective function at every point p of M (where TpX denotes the tangent space of a manifold X at a point p in X). Equivalently, f is an immersion if its derivative has constant rank equal to the dimension of M: The function f itself need not be injective, only its derivative must be. A related concept is that of an embedding.
