Differential calculus

From Exampleproblems

Differential calculus is the theory of and computations with differentials; see also derivative and calculus.

Formally, given a ring-homomorphism of two associative, unitary rings (not necessarily commutative), $R\to A$ , so that $A$ is an $R$ -algebra, a differential $d$ is an $R$ -homomorphism $d\colon A\to A$ satisfying the differential rule $d(a\cdot b) = a\cdot db + (da)\cdot b$ . It follows that $d (R) = 0$ , since applying the differential rule to $d(1\cdot 1)$ gives $\;d1 = 0$ , and $R$ -linearity gives $d(r) = d(r\cdot 1) = r\cdot d1 = 0$ . 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 the space of real numbers, $\mathbf R^n$ , there is a simple differential calculus, which has the advantage of being very practical in explicit computations. Each coordinate $\;x_i$ generates at each point of $\mathbf R^n$ a differential vector $\;dx_i$ , and a list of vectors $(dx_1, \dots, dx_n)$ is linearly independent if and only if the coordinates $x_1, \dots, x_n$ are locally independent. We need only know the following formal relations:

$\; d \mathrm c\; = 0$ for a constant $c$ .
$\; d(\mathrm a f + \mathrm b g) = \mathrm a\;df + \mathrm b\;dg$ for constants $\;\mathrm a, \mathrm b$ and differentiable functions $\;f, g$ .
$\; d(fg) = g df + f dg$ for differentiable functions $\; f$ and $\; g$ .
$df = \frac{\partial f}{\partial x_1} dx_1 + \dots + \frac{\partial f}{\partial x_n} dx_n$ , where $\;f$ is a differentiable function and $x_1, \dots, x_n$ are independent variables on which $\;f$ 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 $\; d$ is coordinate independent, enables practical multidimensional computations with derivatives in a fairly transparent way. For example, suppose we want to compute the derivative $d f / d x$ of the function $\; f$ which depends on the variables $\; x, y$ with the side condition $\; g(x, y) = 0$ . Then one only has to apply $\; d$ to the side condition to get $\; dg = 0$ , which expands to $\; g'_x dx + g'_y dy = 0$ ; if $\; g'_y \neq 0$ , solve for $\; dy$ , and put that into the equation $\; df = f'_x dx + f'_y dy$ , which then becomes equal to $(f'_x + f'_y \frac {g'_x}{g'_y}) dx$ . Thus, $\; \frac{df}{dx} =f'_x + f'_y \frac {g'_x}{g'_y}$ . (Here, $f'_x = \frac{\partial f}{\partial x}$ is the partial derivative where $\; x, y$ move freely, whereas $\; \frac{df}{dx}$ is the derivative of the one-variable function of $\; x$ obtained by substituting the solution of $\; y$ as a function $\; y(x)$ of $\; x$ in the side condition $\; g(x, y) = 0$ .) Similar computations hold in higher dimensions. Without the use of differentials, this kind of computations become tricky.

In the complex space $\mathbf C^n$ , with complex coordinates $\;z_j = x_j + i y_j$ , where $\;i$ is the imaginary unit, the complex differentials are defined as $\; dz_j := dx_j + i dy_j$ and $d{\bar z}_j := dx_j - i dy_j$ . Given a complex valued, real differentiable function $\; f$ , one has $\; df = f'_{x_j} dx_j + f'_{y_j} dy_j = f'_{z_j} dz_j + f'_{{\bar z}_j} d{\bar z}_j$ ; expanding $\; dz_j, d{\bar z}_j$ into $dx_j \pm i dy_j$ , collecting coefficients, solving for $\; f'_{z_j}, f'_{{\bar z}_j}$ , gives the equations, for all $\; j = 1, \dots, n$ :

$f'_{z_j} = \frac{1}{2}(f'_{x_j} - i f'_{y_j})$
$f'_{{\bar z}_j} = \frac{1}{2}(f'_{x_j} + i f'_{y_j})$

A function $\; f$ is called complex differentiable if $f'_{{\bar z}_j} = 0$ for all $\; j = 1, \dots, n$ , and complex analytic in an open set this is true at all points of this set. If $\; f = u + i v$ , then this condition expands to the equations

$u'_{x_j} = v'_{y_j}$
$u'_{y_j} = -v'_{x_j}$

for all $\; j = 1, \dots, n$ . These equations are called the Cauchy-Riemann equations in $\mathbf C^n$ . As the $f'_{z_j}$ are called the holomorphic derivatives of $\; f$ , and $f'_{{\bar z}_j}$ 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 $\mathrm{dlog} f := \frac {df}{f}$ is useful, for example, in the computation of relative errors. The logarithmic differential $\;\mathrm{dlog}$ also shows up in the argument principle of a complex analytic function $\; f(z)$ of one complex variable $\; z$ , which says that path integration along a closed curve $\; C$ of $\;\mathrm{dlog} f$ gives the result $\; 2\pi i(Z-P)$ , where $\; Z$ is the number of zeroes and $\; P$ the number of poles inside $\; C$ , counted with multiplicities and the number of times encircled by $\; C$ (the latter is 1, if $\; C$ is simple and positively oriented).da:Differentialregning de:Differentialrechnung Template:Destacado eo:Derivaĵo es:Cálculo diferencial fr:Dérivée it:Derivata nl:Afgeleide ja:微分 pl:Pochodna funkcji pt:Derivada simple:Derivative fi:Derivaatta sv:Derivata zh-cn:导数