Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/228

This may be easily extended to higher derivatives. For instance, since (58) is true,

${\frac {\partial ^{3}u}{\partial x^{2}\partial y}}={\frac {\partial }{\partial x}}\left({\frac {\partial ^{2}u}{\partial x\partial y}}\right)={\frac {\partial ^{3}u}{\partial x\partial y\partial x}}={\frac {\partial ^{2}}{\partial x\partial y}}\left({\frac {\partial u}{\partial x}}\right)={\frac {\partial ^{2}}{\partial y\partial x}}\left({\frac {\partial u}{\partial x}}\right)={\frac {\partial ^{2}u}{\partial y\partial x^{2}}}.$

Similarly for functions of three or more variables.

Illustrative Example 1. Given $u=x^{3}y-3x^{2}y^{3};$ verify ${\tfrac {\partial ^{2}u}{\partial y\partial x}}={\tfrac {\partial ^{2}u}{\partial x\partial y}}.$

Solution.	${\frac {\partial u}{\partial x}}=3x^{2}y-6xy^{3},{\frac {\partial ^{2}u}{\partial y\partial x}}=3x^{2}-18xy^{2},$
	${\frac {\partial u}{\partial y}}=x^{3}-9x^{2}y^{2},{\frac {\partial ^{2}u}{\partial x\partial y}}=3x^{2}-18xy^{2},$	hence verified.

EXAMPLES

1.	$u=\cos \left(x+y\right);$	verify ${\frac {\partial ^{2}u}{\partial y\partial x}}={\frac {\partial ^{2}u}{\partial x\partial y}}.$
2.	$u={\frac {y^{2}+x^{2}}{y^{2}-x^{2}}};$	verify ${\frac {\partial ^{2}u}{\partial y\partial x}}={\frac {\partial ^{2}u}{\partial x\partial y}}.$
3.	$u=y\log \left(1+xy\right);$	verify ${\frac {\partial ^{2}u}{\partial y\partial x}}={\frac {\partial ^{2}u}{\partial x\partial y}}.$
4.	$u=\arctan {\frac {r}{s}};$	verify ${\frac {\partial ^{3}u}{\partial r^{2}\partial s}}={\frac {\partial ^{3}u}{\partial s\partial r^{2}}}.$
5.	$u=\sin \left(\theta ^{2}\phi \right);$	verify ${\frac {\partial ^{3}u}{\partial \theta \partial \phi ^{2}}}={\frac {\partial ^{3}u}{\partial \phi ^{2}\partial \theta }}.$
6.	$u\ =\ 6e^{x}y^{2}z+3e^{y}x^{2}z^{2}+2e^{z}x^{3}y-xyz;$	show that ${\frac {\partial ^{4}u}{\partial x^{2}\partial y\partial z}}=12\left(e^{x}y+e^{y}z+e^{z}x\right).$
7.	$u\ =\ e^{xyz};$	show that ${\frac {\partial ^{3}u}{\partial x\partial y\partial z}}=\left(1+3xyz+x^{2}y^{2}z^{2}\right)u.$
8.	$u={\frac {x^{2}y^{2}}{x+y}};$	show that $x{\frac {\partial ^{2}u}{\partial x^{2}}}+y{\frac {\partial ^{2}u}{\partial x\partial y}}=2{\frac {\partial u}{\partial x}}.$
9.	$u=\left(x^{2}+y^{2}\right)^{\frac {1}{3}};$	show that $3x{\frac {\partial ^{2}u}{\partial x\partial y}}+3y{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial u}{\partial y}}=0.$
10.	$u=y^{2}z^{2}e^{\frac {x}{2}}+z^{2}x^{2}e^{\frac {y}{2}}+x^{2}y^{2}e^{\frac {z}{2}};$	show that ${\frac {\partial ^{6}u}{\partial x^{2}\partial y^{2}\partial z^{2}}}+3y=e^{\frac {x}{2}}+e^{\frac {y}{2}}+e^{\frac {z}{2}}.$
11.	$u=\left(x^{2}+y^{2}+z^{2}\right)^{-{\frac {1}{2}}};$	show that ${\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.$

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