Differential calculus is the theory of and computations with differentials; see also derivative and calculus. Jump to: navigation, search In mathematics, the derivative is one of the two central concepts of calculus. ... Jump to: navigation, search Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ...

Formally, given a ring-homomorphism of two associative, unitary rings (not necessarily commutative), , so that A is an R-algebra, a differential d is an R-homomorphism satisfying the differential rule . It follows that d(R) = 0, since applying the differential rule to gives , and R-linearity gives . This simple formalism thus identifies R as a set of differential constants.

In the application of differential geometry, the differential will be applied to the ring of smooth functions f, or an extension thereof, via say distributions (or generalized functions). The cotangent bundle generates a differential algebra (an exterior algebra), and differentiation in this algebra leads naturally to de Rham-cohomology, which describes the obstructions of integration. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...

In the space of real numbers, , there is a simple differential calculus, which has the advantage of being very practical in explicit computations. Each coordinate generates at each point of a differential vector , and a list of vectors is linearly independent if and only if the coordinates are locally independent. We need only know the following formal relations:

• for a constant c.
• for constants and differentiable functions .
• for differentiable functions and .
• , where is a differentiable function and are independent variables on which depends (i.e., varies with, and not varying with other variables).

Using these rules, normal multidimensional differentiation can be performed. Integration can be performed by introducing higher dimensional exterior differentials.

The fact that the differential is coordinate independent, enables practical multidimensional computations with derivatives in a fairly transparent way. For example, suppose we want to compute the derivative df / dx of the function which depends on the variables with the side condition . Then one only has to apply to the side condition to get , which expands to ; if , solve for , and put that into the equation , which then becomes equal to . Thus, . (Here, is the partial derivative where move freely, whereas is the derivative of the one-variable function of obtained by substituting the solution of as a function of in the side condition .) Similar computations hold in higher dimensions. Without the use of differentials, this kind of computations become tricky.

In the complex space , with complex coordinates , where is the imaginary unit, the complex differentials are defined as and . Given a complex valued, real differentiable function , one has ; expanding into , collecting coefficients, solving for , gives the equations, for all :

A function is called complex differentiable if for all , and complex analytic in an open set this is true at all points of this set. If , then this condition expands to the equations

for all . These equations are called the Cauchy-Riemann equations in . As the are called the holomorphic derivatives of , and are called the anti-holomorphic derivatives, the Cauchy-Riemann equations just express that the anti-holomorphic derivatives are all zero.

The logarithmic differential defined as is useful, for example, in the computation of relative errors. The logarithmic differential also shows up in the argument principle of a complex analytic function of one complex variable , which says that path integration along a closed curve of gives the result , where is the number of zeroes and the number of poles inside , counted with multiplicities and the number of times encircled by (the latter is 1, if is simple and positively oriented). Template:Destacado