In space regions, points can have variable values and directions {vector field, physics}. For example, points have force and momentum. Scalar-field gradients are vector fields, because gradients have direction and magnitude. Vector fields have gradients, flows, constancies, covariances, contravariances, divergences, curls, and Laplace operators. Total effect of variable over region is vector sum. For example, force-vector sum gives total force.
In space regions, points can have variable values {scalar field, physics}. For example, points have mass density, temperature, and position. Total effect of variable over region is scalar sum. For example, summing mass densities gives total mass.
Vector fields can have complex numbers, instead of real numbers, for vector-component coefficients {spinor field}. Spinor fields require twice the dimension number of corresponding vector fields, because complex numbers have real and imaginary components. Spinor spaces have even number of dimensions.
Moving vector fields can expand outward from points to make waves that superimpose {wave front}|. Wave-front component sums indicate net direction and amplitude.
Differential vector gauge field {Yang-Mills field} for strong and weak nuclear forces can be invariant under transformation {Yang-Mills gauge theory}. Energy increases when reference frame carried around loop does not return to original orientation. Gauge fields can have more than one dimension. Unified field theories require Yang-Mills fields [Yang and Mills, 1954].
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Date Modified: 2022.0225