Partial derivative

In differential calculus , a partial derivative is the derivative of a function with several arguments according to one of these arguments (in the direction of this coordinate axis). The values ​​of the other arguments are therefore kept constant.

definition

First order

Let be an open subset of Euclidean space and a function. Continue to be an element in given. If for the natural number with the limit value ${\displaystyle U}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\colon U\rightarrow \mathbb {R} }$ ${\displaystyle a=(a_{1},\dotsc ,a_{n})}$ ${\displaystyle U}$ ${\displaystyle i}$ ${\displaystyle 1\leq i\leq n}$

${\displaystyle {\frac {\partial f}{\partial x_{i}}}(a):=\lim _{h\to 0}{\frac {f(a_{1},\dotsc ,a_{i}+h,\dotsc ,a_{n})-f(a_{1},\dotsc ,a_{i},\dotsc ,a_{n})}{h}}}$

exists, then it is called the partial derivative of after the -th variable in the point . The function is then called partially differentiable in the point . The symbol ∂ (it is similar to the italic cut of the Cyrillic minuscule д ) is pronounced as or del . The spelling became known through the use of CGJ Jacobi . ${\displaystyle f}$ ${\displaystyle i}$ ${\displaystyle x_{i}}$ ${\displaystyle a}$ ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle d}$ ${\displaystyle {\tfrac {\partial f}{\partial x_{i}}}}$

On the other hand, there is a different notation in technical mechanics in which the direction is displayed with a comma in the index to distinguish it from the direction: The derivation of the displacement (i.e. the displacement in the direction) is equivalent as follows . would be the derivative in -direction of a shift in -direction. ${\displaystyle u_{1}}$ ${\displaystyle x_{1}}$ ${\displaystyle {\frac {\partial u_{1}}{\partial x_{1}}}=u_{1,1}}$ ${\displaystyle {\frac {\partial u_{2}}{\partial x_{3}}}=u_{2,3}}$ ${\displaystyle x_{3}}$ ${\displaystyle x_{2}}$

Higher order

The partial derivative after is itself a function of to , if in is partially differentiable according to . As an abbreviation for the partial derivatives , or is often found. ${\displaystyle x_{i}}$ ${\displaystyle U}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle x_{i}}$ ${\displaystyle {\tfrac {\partial f}{\partial x_{i}}}}$ ${\displaystyle \textstyle \partial _{x_{i}}f}$ ${\displaystyle \textstyle f_{x_{i}}}$ ${\displaystyle D_{i}f}$

If the function is partially differentiable in every point of its domain, then the partial derivatives are ${\displaystyle f\colon U\to \mathbb {R} }$

${\displaystyle {\frac {\partial f}{\partial x_{i}}}\colon a\mapsto {\frac {\partial f}{\partial x_{i}}}(a)}$

again functions from to , which in turn can be examined for differentiability. This gives higher partial derivatives ${\displaystyle U}$ ${\displaystyle \mathbb {R} }$

${\displaystyle {\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}={\frac {\partial }{\partial x_{j}}}\left({\frac {\partial f}{\partial x_{i}}}\right)}$   and   ${\displaystyle \displaystyle {\frac {\partial ^{2}f}{\partial x_{i}^{2}}}={\frac {\partial }{\partial x_{i}}}\left({\frac {\partial f}{\partial x_{i}}}\right)}$

Geometric interpretation

In a three-dimensional coordinate system which is a function graph of a function considered. The domain of definition is an open subset of the xy plane . If differentiable, then the graph of the function is an area over the domain of definition . ${\displaystyle f\colon U\rightarrow \mathbb {R} }$ ${\displaystyle U}$ ${\displaystyle f}$ ${\displaystyle U}$

For a fixed value of then there is a function in . With fixed , the points result in a line parallel to the axis. This distance is projected from on onto a curved line on the graph from . The partial derivation from to corresponds to the slope of the tangent to this curve at the point under these conditions . ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle \{(x,y):y\in \mathbb {R} {\text{ mit }}(x,y)\in U\}}$ ${\displaystyle y}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle y}$ ${\displaystyle f(x,y)}$

Sentences and properties

Connection derivation, partial derivation, continuity

• Totally differentiable functions are continuous.
• Totally differentiable functions are partially differentiable.
• Partially differentiable functions are not necessarily continuous and therefore not necessarily totally differentiable.
• Continuously partially differentiable functions, i.e. functions whose partial derivatives are continuous, are, on the other hand, continuously totally differentiable.

Set of black

• Schwarz's theorem applies : If the second partial derivatives are continuous, the order of the derivative can be reversed:

  ${\displaystyle {\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}={\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}\,.}$

use

• The first partial derivatives can be arranged in a vector, the gradient of : ${\displaystyle f}$

${\displaystyle {\text{grad}}\,f=\nabla f:=\left({\frac {\partial f}{\partial x_{1}}},\ldots ,{\frac {\partial f}{\partial x_{n}}}\right)^{T}}$

Here is the Nabla operator . ${\displaystyle \nabla }$

• The second partial derivatives can be arranged in a matrix, the Hessian matrix

  ${\displaystyle \operatorname {H} _{f}=\left({\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}\right)={\begin{pmatrix}{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{1}}}&\dots &{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{1}}}&\dots &{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{n}}}\end{pmatrix}}}$

• The Taylor formula applies : If the function -mal is continuously partially differentiable, it can be approximated in the vicinity of each point by its Taylor polynomials: ${\displaystyle f\colon U\to \mathbb {R} }$ ${\displaystyle k}$ ${\displaystyle a=(a_{1},\dots ,a_{n})\in U}$

${\displaystyle f(a+h)=\sum _{s=0}^{k}\,\sum _{j_{1}+\dots +j_{n}=s}{\frac {1}{j_{1}!\cdots j_{n}!}}\,{\frac {\partial ^{s}f}{\partial x_{1}^{j_{1}}\cdots \partial x_{n}^{j_{n}}}}(a)\,h_{1}^{j_{1}}\cdots h_{n}^{j_{n}}+r(a,h)}$

with , where the remainder of the term vanishes for of higher than -th order, that is: ${\displaystyle h=(h_{1},\dots ,h_{n})}$ ${\displaystyle r(a,h)}$ ${\displaystyle |h|\to 0}$ ${\displaystyle k}$

${\displaystyle \lim _{|h|\to 0}{\frac {|r(a,h)|}{|h|^{k}}}=0.}$

The terms for a given ν result in the “Taylor approximation -th order”. ${\displaystyle k}$

• Simple extreme value problems can be found in analysis when calculating maxima and minima of a function of a real variable (see also the article on differential calculus ). The generalization of the differential quotient to functions of several variables (variables, parameters) enables the determination of their extreme values, and partial derivatives are required for the calculation.

• In differential geometry , partial derivatives are required to determine a total differential . Applications for total differentials can be found to a large extent in thermodynamics .

• Partial derivations are also an essential part of vector analysis . They form the components of the gradient, the Laplace operator , the divergence and the rotation in scalar and vector fields . They also appear in the Jacobian matrix .

Examples

example 1

The graph of is a paraboloid (animation: GIF export Geogebra ) ${\displaystyle f(x,y):=x^{2}+y^{2}-2}$

As an example, the function is to consider which of the two variables and dependent. ${\displaystyle f\colon \mathbb {R} ^{2}\rightarrow \mathbb {R} }$ ${\displaystyle f(x,y):=x^{2}+y^{2}-2}$ ${\displaystyle x}$ ${\displaystyle y}$

Considered as a constant, e.g. B. , the function with only depends on the variable : ${\displaystyle y}$ ${\displaystyle y=3}$ ${\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }$ ${\displaystyle g(x)=f(x,3)}$ ${\displaystyle x}$

${\displaystyle f(x,3)=x^{2}+7}$

The following applies to the new function and the differential quotient can be calculated ${\displaystyle g(x)=x^{2}+7}$

${\displaystyle {\frac {\mathrm {d} g(x)}{\mathrm {d} x}}=\lim _{h\to 0}{\frac {g(x+h)-g(x)}{h}}=g'(x)=2x}$

The same result is obtained by taking the partial derivative of the function according to : ${\displaystyle f}$ ${\displaystyle x}$

${\displaystyle {\frac {\partial f(x,y)}{\partial x}}=\lim _{h\to 0}{\frac {f(x+h,y)-f(x,y)}{h}}=\lim _{h\to 0}{\frac {(x+h)^{2}+y^{2}-2-x^{2}-y^{2}+2}{h}}=2x}$

The partial derivative of to is accordingly: ${\displaystyle f}$ ${\displaystyle y}$

${\displaystyle {\frac {\partial f(x,y)}{\partial y}}=\lim _{h\to 0}{\frac {f(x,y+h)-f(x,y)}{h}}=\lim _{h\to 0}{\frac {x^{2}+(y+h)^{2}-2-x^{2}-y^{2}+2}{h}}=2y}$

This example demonstrates how to determine the partial derivative of a function that depends on several variables:

With the exception of one variable, all other variables are assumed to be constant; the differential quotient is determined for this one variable. The result is the partial derivative of the function according to this one variable.

Example 2

Since the partial derivative according to a variable corresponds to the normal derivative with fixed values ​​of all other variables, all derivation rules can be used for the calculation as for functions of a variable. Is for example

${\displaystyle f(x,y)=x^{2}\sin(xy)}$,

so it follows with product and chain rule :

${\displaystyle {\frac {\partial f(x,y)}{\partial x}}=2x\sin(xy)+x^{2}y\cos(xy)}$   and

${\displaystyle {\frac {\partial f(x,y)}{\partial y}}=x^{3}\cos(xy)}$.

Example 3

Function plot with Geogebra ${\displaystyle f(x,y):=\cos(x)+\sin(y)}$

In the animation above you can see the graph of the function . If you define a point from the definition range, you can intersect the graph of the function with a vertical plane in the x direction. The intersection of the graph with the plane creates a classic graph from one-dimensional analysis. Partial derivatives can thus also be traced back to the classic one-dimensional analysis. ${\displaystyle f(x,y)=\cos(x)+\sin(y)}$ ${\displaystyle (x_{o},y_{o})\in \mathbb {R} ^{2}}$

${\displaystyle f(x,y)=\cos(x)+\sin(y)}$,

${\displaystyle {\frac {\partial f(x,y)}{\partial x}}=-\sin(x)}$   and

${\displaystyle {\frac {\partial f(x,y)}{\partial y}}=\cos(y)}$.

Partial and total derivative with respect to time

In physics (especially in the theoretical mechanics ) the following situation often occurs: One size hanging by a totally differentiable function of the location coordinates , , and by the time off. One can thus the partial derivatives , , and form. The coordinates of a moving point are given by the functions , and . The time development of the value of the variable at the respective point on the path is then determined by the chained function ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle t}$ ${\displaystyle {\tfrac {\partial f}{\partial x}}}$ ${\displaystyle {\tfrac {\partial f}{\partial y}}}$ ${\displaystyle {\tfrac {\partial f}{\partial z}}}$ ${\displaystyle {\tfrac {\partial f}{\partial t}}}$ ${\displaystyle x(t)}$ ${\displaystyle y(t)}$ ${\displaystyle z(t)}$

${\displaystyle t\mapsto f(x(t),y(t),z(t),t)}$

described. This function only depends on one variable, time . So you can take the usual derivative. This is called the total or complete derivative of after time and is also written briefly for it . It is calculated according to the multidimensional chain rule as follows: ${\displaystyle t}$ ${\displaystyle f}$ ${\displaystyle t}$ ${\displaystyle {\tfrac {\mathrm {d} f}{\mathrm {d} t}}}$

${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}f(x(t),y(t),z(t),t)={\frac {\partial f}{\partial x}}{\frac {\mathrm {d} x}{\mathrm {d} t}}+{\frac {\partial f}{\partial y}}{\frac {\mathrm {d} y}{\mathrm {d} t}}+{\frac {\partial f}{\partial z}}{\frac {\mathrm {d} z}{\mathrm {d} t}}+{\frac {\partial f}{\partial t}}}$

While the partial derivation with respect to time only takes into account the explicit dependence of the function on and all other variables are kept constant, the total derivation also takes into account the indirect (or implicit ) dependence on , which is caused by the spatial coordinates along the path movement depend on the time. ${\displaystyle {\tfrac {\partial f}{\partial t}}}$ ${\displaystyle f}$ ${\displaystyle t}$ ${\displaystyle {\tfrac {\mathrm {d} f}{\mathrm {d} t}}}$ ${\displaystyle t}$

(By taking into account the implicit time dependency, one speaks in the jargon of physics of the "substantial" time derivative, or in the jargon of fluid mechanics of the Euler derivative as opposed to the Lagrange derivative .)

→ For a more detailed representation see total differential

Generalization: Derivation of direction

The directional derivative represents a generalization of the partial derivative . The derivative is considered in the direction of any vector and not just in the direction of the coordinate axes.

literature

• Kurt Endl; Wolfgang Luh: Analysis II , Akademische Verlagsgesellschaft Frankfurt am Main, 1974
• Hans Grauert; Wolfgang Fischer: Differential- und Integralrechnung II , 2nd, improved edition, Springer Verlag Berlin, 1978

Individual evidence

1. ↑ Heuser refers to J. f. pure u. Applied Math. , No. 17 (1837) (Harro Heuser: Textbook of Analysis. Part 2. , Teubner Verlag, 2002, p. 247). Jeff Miller gives a detailed origin: [1] .
2. ^ Holm Altenbach, Johannes Altenbach, Konstantin Naumenko: level surface structures . Basics of modeling and calculation of panes and plates. Springer, Berlin Heidelberg 2016, ISBN 978-3-662-47230-9 , pp. 25th ff ., doi : 10.1007 / 978-3-662-47230-9 .