Differentials of higher orders

The differential of order n , where n> 1 , from the function $z$ ${\ displaystyle z}$ $z$ at some point is called the differential at this point from the differential of order (n - 1) , that is,

d^{n}z=d(d^{n-1}z)

{\ displaystyle d ^ {n} z = d (d ^ {n-1} z)}

{\ displaystyle d ^ {n} z = d (d ^ {n-1} z)}

.

Content

Higher order differential of a single variable function

For a function dependent on one independent variable $z=f(x)$ ${\ displaystyle z = f (x)}$ The second and third differentials look like this:

d^{2}z=d(dz)=d(z'dx)=dz'dx=(z''dx)dx=z''dx^{2}

{\ displaystyle d ^ {2} z = d (dz) = d (z'dx) = dz'dx = (z''dx) dx = z''dx ^ {2}}

,

d^{3}z=d(d^{2}z)=d(z''dx^{2})=dz''dx^{2}=(z'''dx)dx^{2}=z'''dx^{3}

{\ displaystyle d ^ {3} z = d (d ^ {2} z) = d (z''dx ^ {2}) = dz''dx ^ {2} = (z '' dx) dx ^ {2} = z '' 'dx ^ {3}}

.

From here you can deduce the general form of the differential of the nth order from the function $z=f(x)$ ${\ displaystyle z = f (x)}$ , provided that $x$ ${\ displaystyle x}$ - independent variable:

d^{n}z=z^{(n)}dx^{n}

{\ displaystyle d ^ {n} z = z ^ {(n)} dx ^ {n}}

.

When calculating differentials of higher orders, it is very important that $d x$ ${\ displaystyle dx}$ there is arbitrary and independent of $x$ ${\ displaystyle x}$ which when differentiating by $x$ ${\ displaystyle x}$ should be considered as a constant multiplier. If a $x$ ${\ displaystyle x}$ is not an independent variable, the differential will be different (see below ) ^[1] .

Higher-order differential functions of several variables

If the function $z=f(x,y)$ ${\ displaystyle z = f (x, y)}$ has continuous partial derivatives of the second order, then the differential of the second order is defined as: $d^{2}z=d(dz)$ ${\ displaystyle d ^ {2} z = d (dz)}$ .

d^{2}z=d\left({\frac {\partial z}{\partial x}}dx+{\frac {\partial z}{\partial y}}dy\right)=\left({\frac {\partial z}{\partial x}}dx+{\frac {\partial z}{\partial y}}dy\right)'_{x}dx+\left({\frac {\partial z}{\partial x}}dx+{\frac {\partial z}{\partial y}}dy\right)'_{y}dy=

{\ displaystyle d ^ {2} z = d \ left ({\ frac {\ partial z} {\ partial x}} dx + {\ frac {\ partial z} {\ partial y}} dy \ right) = \ left ({\ frac {\ partial z} {\ partial x}} dx + {\ frac {\ partial z} {\ partial y}} dy \ right) '_ {x} dx + \ left ({\ frac {\ partial z } {\ partial x}} dx + {\ frac {\ partial z} {\ partial y}} dy \ right) '_ {y} dy =}

=\left({\frac {\partial ^{2}z}{\partial x^{2}}}dx+{\frac {\partial ^{2}z}{\partial y\partial x}}dy\right)dx+\left({\frac {\partial ^{2}z}{\partial x\partial y}}dx+{\frac {\partial ^{2}z}{\partial y^{2}}}dy\right)dy

{\ displaystyle = \ left ({\ frac {\ partial ^ {2} z} {\ partial x ^ {2}}} dx + {\ frac {\ partial ^ {2} z} {\ partial y \ partial x} } dy \ right) dx + \ left ({\ frac {\ partial ^ {2} z} {\ partial x \ partial y}} dx + {\ frac {\ partial ^ {2} z} {\ partial y ^ {2 }}} dy \ right) dy}

d^{2}z={\frac {\partial ^{2}z}{\partial x^{2}}}dx^{2}+2{\frac {\partial ^{2}z}{\partial x\partial y}}dxdy+{\frac {\partial ^{2}z}{\partial y^{2}}}dy^{2}

{\ displaystyle d ^ {2} z = {\ frac {\ partial ^ {2} z} {\ partial x ^ {2}}} dx ^ {2} +2 {\ frac {\ partial ^ {2} z } {\ partial x \ partial y}} dxdy + {\ frac {\ partial ^ {2} z} {\ partial y ^ {2}} dy ^ {2}}

d^{2}z=\left({\frac {\partial }{\partial x}}dx+{\frac {\partial }{\partial y}}dy\right)^{2}z

{\ displaystyle d ^ {2} z = \ left ({\ frac {\ partial} {\ partial x}} dx + {\ frac {\ partial} {\ partial y}} dy \ right) ^ {2} z}

Symbolically, the general form of the differential of the nth order of the function $z=f(x_{1},...,x_{r})$ ${\ displaystyle z = f (x_ {1}, ..., x_ {r})}$ as follows:

d^{n}z=\left({\frac {\partial }{\partial x_{1}}}dx_{1}+{\frac {\partial }{\partial x_{2}}}dx_{2}+...+{\frac {\partial }{\partial x_{r}}}dx_{r}\right)^{n}z

{\ displaystyle d ^ {n} z = \ left ({\ frac {\ partial} {\ partial x_ {1}}} dx_ {1} + {\ frac {\ partial} {\ partial x_ {2}}} dx_ {2} + ... + {\ frac {\ partial} {\ partial x_ {r}}} dx_ {r} \ right) ^ {n} z}

Where $z=f(x_{1},x_{2},...x_{r})$ ${\ displaystyle z = f (x_ {1}, x_ {2}, ... x_ {r})}$ , but $dx_{1},...,dx_{r}$ ${\ displaystyle dx_ {1}, ..., dx_ {r}}$ arbitrary increments of independent variables $x_{1},...,x_{r}$ ${\ displaystyle x_ {1}, ..., x_ {r}}$ .
Increments $dx_{1},...,dx_{r}$ ${\ displaystyle dx_ {1}, ..., dx_ {r}}$ considered as constant and remain the same when moving from one differential to the next. The complexity of expressing the differential increases with the number of variables.

Noninvariance of higher order differentials

With $n\geqslant 2$ ${\ displaystyle n \ geqslant 2}$ $n$ ${\ displaystyle n}$ differential is not invariant (as opposed to the invariance of the first differential ), that is, the expression $d^{n}f$ ${\ displaystyle d ^ {n} f}$ depends, generally speaking, on whether a variable is considered $x$ ${\ displaystyle x}$ as independent, or as some intermediate function of another variable, for example, $x=\varphi (t)$ ${\ displaystyle x = \ varphi (t)}$ .

So, for the independent variable $x$ ${\ displaystyle x}$ the second differential, as mentioned above, has the form:

d^{2}z=z''(dx)^{2}

{\ displaystyle d ^ {2} z = z '' (dx) ^ {2}}

If variable $x$ ${\ displaystyle x}$ itself may depend on other variables then $d(dx)=d^{2}x\neq 0$ ${\ displaystyle d (dx) = d ^ {2} x \ neq 0}$ . In this case, the formula for the second differential will have the form ^[1] :

d^{2}z=d(dz)=d(z'dx)=z''\,(dx)^{2}+z'd^{2}x

{\ displaystyle d ^ {2} z = d (dz) = d (z'dx) = z '' \, (dx) ^ {2} + z'd ^ {2} x}

.

Similarly, the third differential will take the form:

d^{3}z=z'''\,(dx)^{3}+3z''dx\,d^{2}x+z'd^{3}x

{\ displaystyle d ^ {3} z = z '' '\, (dx) ^ {3} + 3z''dx \, d ^ {2} x + z'd ^ {3} x}

.

To prove the non-invariance of higher order differentials, it suffices to give an example.
With $n=2$ ${\ displaystyle n = 2}$ and $y=f(x)=x^{3}$ ${\ displaystyle y = f (x) = x ^ {3}}$ :

if a $x$ ${\ displaystyle x}$ - independent variable, then $d^{2}y=d^{2}f(x)=(x^{3})''(dx)^{2}=6x(dx)^{2}$ ${\ displaystyle d ^ {2} y = d ^ {2} f (x) = (x ^ {3}) '' (dx) ^ {2} = 6x (dx) ^ {2}}$
if a $x$ $=$ $φ$ $($ $t$ $)$ $=$ $t$ $2$ {\ displaystyle x = \ varphi (t) = t ^ {2}} and $d$ $x$ $=$ $d$ $φ$ $($ $t$ $)$ $=$ $φ$ $'$ $($ $t$ $)$ $d$ $t$ $=$ $2$ $t$ $d$ $t$ {\ displaystyle dx = d \ varphi (t) = \ varphi '(t) dt = 2tdt}
1. $6x(dx)^{2}=6t^{2}(2tdt)^{2}=\color {BrickRed}{24t^{4}(dt)^{2}}$ ${\ displaystyle 6x (dx) ^ {2} = 6t ^ {2} (2tdt) ^ {2} = \ color {BrickRed} {24t ^ {4} (dt) ^ {2}}}$
2. wherein, $y=x^{3}=(t^{2})^{3}=t^{6}$ ${\ displaystyle y = x ^ {3} = (t ^ {2}) ^ {3} = t ^ {6}}$ and $d^{2}y=(t^{6})''(dt)^{2}=\color {BrickRed}{30t^{4}(dt)^{2}}$ ${\ displaystyle d ^ {2} y = (t ^ {6}) '' (dt) ^ {2} = \ color {BrickRed} {30t ^ {4} (dt) ^ {2}}}$

In view of dependence $x=t^{2}$ ${\ displaystyle x = t ^ {2}}$ , already the second differential does not possess the property of invariance when changing a variable. The differentials of orders 3 and higher are also not invariant.

Additions

Using differentials function $F$ ${\ displaystyle F}$ subject to its existence $(n+1)$ ${\ displaystyle (n + 1)}$ first derivatives can be represented by Taylor formula :

for a function with one variable:

{\mathcal {4}}F(x_{0})=dF(x_{0})+{\frac {d^{2}F(x_{0})}{2!}}+...+{\frac {d^{n}F(x_{0})}{n!}}+{\frac {d^{n+1}F(x_{0}+\theta {\mathcal {4}}x)}{(n+1)!}}

{\ displaystyle {\ mathcal {4}} F (x_ {0}) = dF (x_ {0}) + {\ frac {d ^ {2} F (x_ {0})} {2!}} +. .. + {\ frac {d ^ {n} F (x_ {0})} {n!}} + {\ frac {d ^ {n + 1} F (x_ {0} + \ theta {\ mathcal { 4}} x)} {(n + 1)!}}}

,

(0<\theta <1)

{\ displaystyle (0 <\ theta <1)}

;

for a function with several variables:

{\mathcal {4}}F(x_{0},y_{0})=dF(x_{0},y_{0})+{\frac {d^{2}F(x_{0},y_{0})}{2!}}+...+{\frac {d^{n}F(x_{0},y_{0})}{n!}}+{\frac {d^{n+1}F(x_{0}+\theta {\mathcal {4}}x,y_{0}+\theta {\mathcal {4}}y)}{(n+1)!}}

{\ displaystyle {\ mathcal {4}} F (x_ {0}, y_ {0}) = dF (x_ {0}, y_ {0}) + {\ frac {d ^ {2} F (x_ {0 }, y_ {0})} {2!}} + ... + {\ frac {d ^ {n} F (x_ {0}, y_ {0})} {n!}} + {\ frac { d ^ {n + 1} f (x_ {0} + \ theta {\ mathcal {4}} x, y_ {0} + \ theta {\ mathcal {4}} y)} {(n + 1)!} }}

,

(0<\theta <1)

{\ displaystyle (0 <\ theta <1)}

If the first differential is zero, and the second differential of the function $f(x_{1},...,x_{n})$ ${\ displaystyle f (x_ {1}, ..., x_ {n})}$ is positively defined (negatively defined), then the point $(x_{1},...,x_{n})$ ${\ displaystyle (x_ {1}, ..., x_ {n})}$ is a point of strict minimum (according to strict maximum); if the second differential of the function $f(x_{1},...,x_{n})$ ${\ displaystyle f (x_ {1}, ..., x_ {n})}$ is uncertain then at the point $(x_{1},...,x_{n})$ ${\ displaystyle (x_ {1}, ..., x_ {n})}$ no extremum .

Notes

↑ ¹ ² Baranova Elena Semenovna, Vasilyeva Natalya Viktorovna, Fedotov Valery Pavlovich. Practical manual on higher mathematics. Model calculations: Tutorial. 2nd ed. . - "Publishing House" "Peter" "", 2012. - p. 196-197. - 400 s. - ISBN 9785496000123 .

Literature

G.M. Fichtengolts "Course of differential and integral calculus", volume 1

[:0-1] ¹ ² Baranova Elena Semenovna, Vasilyeva Natalya Viktorovna, Fedotov Valery Pavlovich. Practical manual on higher mathematics. Model calculations: Tutorial. 2nd ed. . - "Publishing House" "Peter" "", 2012. - p. 196-197. - 400 s. - ISBN 9785496000123 .