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According to problem 25 in Kühnel's "Differential Geometry Curves – Surfaces – Manifolds", it is also true that two Bertrand curves that do not lie in the same two-dimensional plane are characterized by the existence of a linear relation a κ(t) + b τ(t) = 1 where κ(t) and τ(t) are the curvature and torsion of γ 1 (t) and a and b are ...
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 mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
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.
Such equations give rise to the terminology found in some texts wherein the derivative is referred to as the "differential coefficient" (i.e., the coefficient of dx). Some authors and journals set the differential symbol d in roman type instead of italic: dx. The ISO/IEC 80000 scientific style guide recommends this style.
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.
In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when
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 ).