Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/217

Similarly, if we pass the plane BCD through P parallel to the YOZ-plane, its equation is $x=a,$

and for the section $DPI,{\tfrac {\partial z}{\partial y}}$ means the same as ${\tfrac {dz}{dy}}$. Hence

${\frac {\partial z}{\partial y}}={\frac {dz}{dy}}=-\tan MT'P=$ slope of section DI at P.

Illustrative Example 1. Given the ellipsoid ${\frac {x^{2}}{24}}+{\frac {y^{2}}{12}}+{\frac {z^{2}}{6}}=1$; find the slope of the section of the ellipsoid made (a) by the plane $y=1$ at the point where $x=4$ and $z$ is positive; (b) by the plane $x=2$ at the point where $y=3$ and $z$ is positive.

Solution.	Considering $y$ as constant,
	${\tfrac {2x}{24}}+{\tfrac {2z}{6}}{\tfrac {\partial z}{\partial x}}$	= 0, or ${\tfrac {\partial z}{\partial x}}=-{\tfrac {x}{4z}}.$
When $x$ is a constant	${\tfrac {2y}{12}}+{\tfrac {2z}{6}}{\tfrac {\partial z}{\partial y}}$	= 0, or ${\tfrac {\partial z}{\partial y}}=-{\tfrac {y}{2z}}.$

(a) When $y=1$ and $x=4,z={\sqrt {\tfrac {3}{2}}}$. ∴ ${\tfrac {\partial z}{\partial x}}=-{\sqrt {\tfrac {2}{3}}}.$ Ans.

(b) When $x=2$ and $y=3,z={\tfrac {1}{\sqrt {2}}}$. ∴ ${\tfrac {\partial z}{\partial y}}=-{\tfrac {3}{2}}{\sqrt {2}}.$ Ans.

EXAMPLES

1.	$u=x^{3}+3x^{2}y-y^{3}.$	Ans.	${\tfrac {\partial u}{\partial x}}=3x^{2}+6xy;$
			${\tfrac {\partial u}{\partial y}}=3x^{2}+3y^{2}.$
2.	$u=Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F.$		${\tfrac {\partial u}{\partial x}}=2Ax+By+d;$
			${\tfrac {\partial u}{\partial y}}=Bx+2Cy+E.$
3.	$u=(ax^{2}+by^{2}+cz^{2})^{n}.$		${\tfrac {\partial u}{\partial x}}={\frac {2anxu}{ax^{2}+by^{2}+cz^{2}}};$
			${\tfrac {\partial u}{\partial y}}={\frac {2bnxu}{ax^{2}+by^{2}+cz^{2}}}$
4.	$u=\arcsin {\tfrac {x}{y}}.$		${\tfrac {\partial u}{\partial x}}={\frac {1}{\sqrt {y^{2}-x^{2}}}};$
			${\tfrac {\partial u}{\partial y}}=-{\frac {x}{y{\sqrt {y^{2}-x^{2}}}}}.$
5.	$u=x^{y}.$		${\tfrac {\partial u}{\partial x}}=yx^{y-1};$
			${\tfrac {\partial u}{\partial y}}=x^{y}\log x.$
6.	$u=ax^{3}y^{2}z+bxy^{3}z^{4}+cy^{6}+dxz^{3}.$		${\tfrac {\partial u}{\partial x}}=3ax^{2}y^{2}z+by^{3}z^{4}+dz^{3}.$
			${\tfrac {\partial u}{\partial y}}=2ax^{3}yz+3bxy^{2}z^{4}+6cy^{5}.$
			${\tfrac {\partial u}{\partial z}}=ax^{3}y^{2}+4bxy^{3}z^{3}+3dxz^{2}.$