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The symbol was introduced originally in 1770 by Nicolas de Condorcet, who used it for a partial differential, and adopted for the partial derivative by Adrien-Marie Legendre in 1786. [3] It represents a specialized cursive type of the letter d , just as the integral sign originates as a specialized type of a long s (first used in print by ...
Partial derivatives are generally distinguished from ordinary derivatives by replacing the differential operator d with a "∂" symbol. For example, we can indicate the partial derivative of f ( x , y , z ) with respect to x , but not to y or z in several ways:
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:
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.
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 ...
The exterior derivative is an operation on differential forms that, given a k-form , produces a (k+1)-form . This operation extends the differential of a function (a function can be considered as a 0 -form, and its differential is d f ( x ) = f ′ ( x ) d x {\displaystyle df(x)=f'(x)dx} ).
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
For an n-manifold M, the Hodge star operator: () is a duality mapping taking a -form () to an ()-form () (). It can be defined in terms of an oriented frame ( X 1 , … , X n ) {\displaystyle (X_{1},\ldots ,X_{n})} for T M {\displaystyle TM} , orthonormal with respect to the given metric tensor g {\displaystyle g} :