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Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the "time derivative" — the rate of change over time — is essential for the precise ...
In physics this theorem is one of the ways of defining a conservative force. By placing φ as potential, ∇ φ is a conservative field . Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows.
A parametric C r-curve or a C r-parametrization is a vector-valued function: that is r-times continuously differentiable (that is, the component functions of γ are continuously differentiable), where , {}, and I is a non-empty interval of real numbers.
In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one.
For example, suppose we want to find the integral . Since this is a product of two functions that are simple to integrate separately, repeated integration by parts is certainly one way to evaluate it.
Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. [1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
Usually, the existence of multiple solutions is not a problem, since functions are considered to be equivalent in the theory of L p spaces and Sobolev spaces if they are equal almost everywhere. The characteristic function of the rational numbers 1 Q {\displaystyle 1_{\mathbb {Q} }} is nowhere differentiable yet has a weak derivative.