Partial Differentiation

We sometimes come across quantities that are functions of more than one independent variable. Thus, we may find a case where $y$ depends on two other variable quantities, one of which we will call $u$ and the other $v$. In symbols \[ dy = \frac{\partial y}{\partial u}\, du + \dfrac{\partial y}{\partial v}\, dv; \] \[ y = f(u, v). \] Take the simplest concrete case. Let \[ y = u×v. \] What are we to do? If we were to treat $v$ as a constant, and differentiate with respect to $u$, we should get \[ dy_v = v\, du; \] or if we treat $u$ as a constant, and differentiate with respect to $v$, we should have: \[ dy_u = u\, dv. \]

which are *partial differentials*.

* Example (1)* Find the partial differential coefficients
of the expression $w = 2ax^2 + 3bxy + 4cy^3$.
The answers are:
\[
\left.
\begin{aligned}
\frac{\partial w}{\partial x} &= 4ax + 3by. \\
\frac{\partial w}{\partial y} &= 3bx + 12cy^2.
\end{aligned} \right\}
\]

* Example (2)* Let $z = x^y$. Then, treating first $y$
and then $x$ as constant, we get in the usual way
\[
\left.
\begin{aligned}
\dfrac{\partial z}{\partial x} &= yx^{y-1}, \\
\dfrac{\partial z}{\partial y} &= x^y × \log_\epsilon x,
\end{aligned}\right\}
\]
so that $dz = yx^{y-1}\, dx + x^y \log_\epsilon x \, dy$.

* Example (3)* A cone having height $h$ and radius
of base $r$ has volume $V=\frac{1}{3} \pi r^2 h$. If its height remains
constant, while $r$ changes, the ratio of change of
volume, with respect to radius, is different from ratio
of change of volume with respect to height which
would occur if the height were varied and the radius
kept constant, for
\[
\left.
\begin{aligned}
\frac{\partial V}{\partial r} &= \dfrac{2\pi}{3} rh, \\
\frac{\partial V}{\partial h} &= \dfrac{\pi}{3} r^2.
\end{aligned}\right\}
\]

* Example (4)* In the following example $F$ and $f$
denote two arbitrary functions of any form whatsoever.
For example, they may be sine-functions, or
exponentials, or mere algebraic functions of the two
independent variables, $t$ and $x$. This being understood,
let us take the expression
\begin{align*}
y &= F(x+at) + f(x-at), \\
\text{or,}\;\quad y &= F(w) + f(v); \\
\text{where}\;\quad w &= x+at,\quad \text{and}\quad v = x-at. \\
\text{Then}\;\quad \frac{\partial y}{\partial x}
&= \frac{\partial F(w)}{\partial w} · \frac{\partial w}{\partial x}
+ \frac{\partial f(v)}{\partial v} · \frac{\partial v}{\partial x} \\
&= F'(w) · 1 + f'(v) · 1
\end{align*}
(where the figure $1$ is simply the coefficient of $x$ in
$w$ and $v$)
\begin{align*}
\text{and}\; \ \frac{\partial^2 y}{\partial x^2}
&= F''(w) + f''(v). && \\
\text{Also}\; \ \frac{\partial y}{\partial t}
&= \frac{\partial F(w)}{\partial w} · \frac{\partial w}{\partial t}
+ \frac{\partial f(v)}{\partial v} · \frac{\partial v}{\partial t} \\
&= F'(w) · a - f'(v) a; \\
\text{and}\; \ \frac{\partial^2 y}{\partial t^2}
&= F''(w)a^2 + f''(v)a^2; \\
\text{whence}\; \ \frac{\partial^2 y}{\partial t^2}
&= a^2\, \frac{\partial^2 y}{\partial x^2}.
\end{align*}

This differential equation is of immense importance in mathematical physics.

* Example (5)* Let us take up again Exercise IX. No. 4.

An immediate solution is $x=y$.

* Example (6)* Find the dimensions of an ordinary
railway coal truck with rectangular ends, so that,
for a given volume $V$ the area of sides and floor
together is as small as possible.

(2) Find the partial differential coefficients with respect to $x$, $y$ and $z$, of the expression \[ x^2yz + xy^2z + xyz^2 + x^2y^2z^2. \]

(3) Let $r^2 = (x-a)^2 + (y-b)^2 + (z-c)^2$.

Find the value of $\dfrac{\partial r}{\partial x} + \dfrac{\partial r}{\partial y} + \dfrac{\partial r}{\partial z}$. Also find the value of $\dfrac{\partial^2r}{\partial x^2} + \dfrac{\partial^2r}{\partial y^2} + \dfrac{\partial^2r}{\partial z^2}$.

(4) Find the total differential of $y=u^v$.

(5) Find the total differential of $y=u^3 \sin v$; of $y = (\sin x)^u$; and of $y = \dfrac{\log_\epsilon u}{v}$.

(6) Verify that the sum of three quantities $x$, $y$, $z$, whose product is a constant $k$, is maximum when these three quantities are equal.

(7) Find the maximum or minimum of the function \[ u = x + 2xy + y. \]

(8) The post-office regulations state that no parcel
is to be of such a size that its length plus its girth
exceeds $6$ feet. What is the greatest volume that
can be sent by post (*a* ) in the case of a package of
rectangular cross section; (*b* ) in the case of a package
of circular cross section.

(9) Divide $\pi$ into $3$ parts such that the continued product of their sines may be a maximum or minimum.

(10) Find the maximum or minimum of $u = \dfrac{\epsilon^{x+y}}{xy}$.

(11) Find maximum and minimum of \[ u = y + 2x - 2 \log_\epsilon y - \log_\epsilon x. \]

(12) A telpherage bucket of given capacity has the shape of a horizontal isosceles triangular prism with the apex underneath, and the opposite face open. Find its dimensions in order that the least amount of iron sheet may be used in its construction.

(1) $x^3 - 6x^2 y - 2y^2;\quad \frac{1}{3} - 2x^3 - 4xy$.

(2) $2xyz + y^2 z + z^2 y + 2xy^2 z^2$;

$2xyz + x^2 z + xz^2 + 2x^2 yz^2$;

$2xyz + x^2 y + xy^2 + 2x^2 y^2 z$.

(3) $\dfrac{1}{r} \{ \left(x - a\right) + \left( y - b \right) + \left( z - c \right) \} = \dfrac{ \left( x + y + z \right) - \left( a + b + c \right) }{r}$; $\dfrac{3}{r}$.

(4) $dy = vu^{v-1}\, du + u^v \log_\epsilon u\, dv$.

(5) $dy = 3\sin v u^2\, du + u^3 \cos v\, dv$,

$dy = u \sin x^{u-1} \cos x\, dx + (\sin x)^u \log_\epsilon \sin x du$,

$dy = \dfrac{1}{v}\, \dfrac{1}{u}\, du - \log_\epsilon u \dfrac{1}{v^2}\, dv$.

(7) Minimum for $x = y = -\frac{1}{2}$.

(8) (*a* ) Length $2$ feet, width = depth = $1$ foot, vol. = $2$ cubic
feet.
(*b* ) Radius = $\dfrac{2}{\pi}$ feet = $7.46$ in., length = $2$ feet, vol. = $2.54$.

(9) All three parts equal; the product is maximum.

(10) Minimum for $x = y = 1$.

(11) Min.: $x = \frac{1}{2}$ and $y = 2$.

(12) Angle at apex $= 90°$; equal sides = length = $\sqrt[3]{2V}$.

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