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In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. [1] It is one of the two traditional divisions of calculus, the other being integral calculus —the study of the area beneath a curve.
The linear map is called the (total) derivative or (total) differential of at . Other notations for the total derivative include D a f {\displaystyle D_{a}f} and D f ( a ) {\displaystyle Df(a)} . A function is ( totally ) differentiable if its total derivative exists at every point in its domain.
The resulting calculus, known as exterior calculus, allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential k-form is thought of as measuring the flux through an infinitesimal k-parallelotope at each point of the manifold, then its exterior ...
The step size is =. The same illustration for = The midpoint method converges faster than the Euler method, as .. Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs).
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.
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach .
In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point.
In geometric calculus, the geometric derivative satisfies a weaker form of the Leibniz (product) rule. It specializes the Fréchet derivative to the objects of geometric algebra. Geometric calculus is a powerful formalism that has been shown to encompass the similar frameworks of differential forms and differential geometry. [1]
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