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which, one hopes to prove, is the Taylor series of the desired solution. The Taylor series may also be generalized to functions of more than one variable...

formulas for the remainder term (given below) which are valid under some additional regularity assumptions on f. These enhanced versions of Taylor's theorem...

theory of generalized functions in order to define weak solutions of partial differential equations (i.e. solutions which are generalized functions,...

higher dimensions. A Taylor series is a generalization of a MacLaurin series. The binomial formula is a generalization of the formula for ( 1 + x ) n {\displaystyle...

{2}{k}}f^{(2-k)}(x)g^{(k)}(x)}=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x).} The formula can be generalized to the product of m differentiable functions f1,...,fm. ( f 1...

integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused...

Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function...

complexity Asymptotic expansion: Approximation of functions generalizing Taylor's formula Asymptotically optimal algorithm: A phrase frequently used to...

In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called...

Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno (1855...

The vector calculus operations of grad, curl, and div are most easily generalized in the context of differential forms, which involves a number of steps...

\end{bmatrix}}.} This matrix is skew-symmetric. The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by Camille Jordan in 1874...

theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously...

coefficients with such first arguments. These "generalized binomial coefficients" appear in Newton's generalized binomial theorem. For each k, the polynomial...

in 1769. Although generalized to triple integrals by Lagrange in 1773, and used by Legendre, Laplace, and Gauss, and first generalized to n variables by...

inner point, hence the above limits exist and are real numbers. This generalized version of the theorem is sufficient to prove convexity when the one-sided...

calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For...

also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similar formula. Another generalization...

and its coefficients can be found by a generalization of the above formulas. Taylor's theorem gives a precise bound on how good the approximation is. If...

In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives...