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Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
Given a differentiable manifold, a vector field on is an assignment of a tangent vector to each point in . [2] More precisely, a vector field F {\displaystyle F} is a mapping from M {\displaystyle M} into the tangent bundle T M {\displaystyle TM} so that p ∘ F {\displaystyle p\circ F} is the identity mapping where p {\displaystyle p} denotes ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
The space of solutions is the affine subspace x + V where x is a particular solution of the equation, and V is the space of solutions of the homogeneous equation (the nullspace of A). The set of one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space ; it may be used to formalize the idea of parallel ...
For example, [5] suppose that we are given a basis e 1, e 2 consisting of a pair of vectors making a 45° angle with one another, such that e 1 has length 2 and e 2 has length 1. Then the dual basis vectors are given as follows: e 2 is the result of rotating e 1 through an angle of 90° (where the sense is measured by assuming the pair e 1, e 2 ...
The second point is randomly chosen in the same cube. If the angle between the vectors was within π/2 ± 0.037π/2 then the vector was retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within π/2 ± 0.037π/2 then the vector is ...
IUPAC definition for a crosslink in polymer chemistry In chemistry and biology , a cross-link is a bond or a short sequence of bonds that links one polymer chain to another. These links may take the form of covalent bonds or ionic bonds and the polymers can be either synthetic polymers or natural polymers (such as proteins ).