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Let be an -grade multivector.Then we can define an additional pair of operators, the interior and exterior derivatives, = =, = + =. In particular, if is grade 1 (vector-valued function), then we can write
If the direction of derivative is not repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C 2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:
If all second-order partial derivatives of exist, then the Hessian matrix of is a square matrix, usually defined and arranged as = []. That is, the entry of the i th row and the j th column is ( H f ) i , j = ∂ 2 f ∂ x i ∂ x j . {\displaystyle (\mathbf {H} _{f})_{i,j}={\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}.}
The derivative of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of formal integration by parts to define differentiation of distributions puts the symmetry question back onto the test functions , which are smooth and certainly ...
[3] The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. [1] [2]: 6 Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
Separation of variables may be possible in some coordinate systems but not others, [2] and which coordinate systems allow for separation depends on the symmetry properties of the equation. [3] Below is an outline of an argument demonstrating the applicability of the method to certain linear equations, although the precise method may differ in ...
Combining derivatives of different variables results in a notion of a partial differential operator. The linear operator which assigns to each function its derivative is an example of a differential operator on a function space. By means of the Fourier transform, pseudo-differential operators can be defined which allow for fractional calculus.
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