Scalar vs Vector

A scalar has magnitude only: a single number with units. Examples: mass (kg), temperature (K), energy (J), electric potential (V), time (s), charge (C).

A vector has both magnitude and direction. In 3-D, a vector has three components transforming together under rotations. Examples: velocity (m/s with direction), force (N), momentum (kg·m/s), electric field (V/m), acceleration (m/s²).

Scalars add by ordinary arithmetic. Vectors add component-wise, equivalently by the parallelogram rule. The dot product of two vectors is a scalar; the cross product (in 3-D) is a vector. Higher-rank objects — tensors — generalise scalars (rank-0) and vectors (rank-1) and appear throughout relativity and continuum mechanics.

Recent research on this topic from arXiv

Preprints and papers indexed on arXiv.org. Links open the public abstract pages.

General Fractional Vector Calculus
Vasily E. Tarasov · 2021 · arXiv:2111.02716v1
A generalization of fractional vector calculus as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency in...
A Tensor Rank Theory and Maximum Full Rank Subtensors
Liqun Qi, Xinzhen Zhang, Yannan Chen · 2020 · arXiv:2004.11240v7
A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix prope...
Non-minimum tensor rank Gabidulin codes
Daniele Bartoli, Giovanni Zini, Ferdinando Zullo · 2022 · arXiv:2201.08242v1
The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an $8$-dimensional $\mathbb{F}_q$-linear generalized Gabidulin code in $\mathbb{F}...
Nonlocal vector calculus on the sphere
Hadrien Montanelli, Richard Mikael Slevinsky, Qiang Du · 2025 · arXiv:2505.12372v1
We introduce a nonlocal vector calculus on the unit two-sphere using weakly singular integral operators. Within this framework, the operators are diagonalizable in terms of scalar and vector spherical harmonics, a property that facilitates...

Recent research on this topic from arXiv

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