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Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
This states that differentiation is the reverse process to integration. Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration.
The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.
The 4-wavevector is the 4-gradient of the negative phase (or the negative 4-gradient of the phase) of a wave in Minkowski Space: [6]: 387 = = (,) = [] = [] This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave ): K ⋅ X = ω t − k → ⋅ x → = − Φ {\displaystyle \mathbf {K} \cdot ...
where the f ab are formed from the electromagnetic fields and ; e.g., f 12 = E z /c, f 23 = −B z, or equivalent definitions. This form is a special case of the curvature form on the U(1) principal bundle on which both electromagnetism and general gauge theories may be described.
The exterior derivative is a notion of differentiation of differential forms which generalizes the differential of a function (which is a differential 1-form). Pullback is, in particular, a geometric name for the chain rule for composing a map between manifolds with a differential form on the target manifold.
Isaac Newton's notation for differentiation (also called the dot notation, fluxions, or sometimes, crudely, the flyspeck notation [12] for differentiation) places a dot over the dependent variable. That is, if y is a function of t, then the derivative of y with respect to t is
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.