Partial Differentiation

Excerpts

Partial Differentiation
The first is obtained by supposing $y$ constant, the second is obtained by supposing $x$ constant; then
Partial Differentiation
Clearly $A$ is maximum when $P$ is maximum.
Partial Differentiation
The variation when both the radius and the height change is given by $dV = \dfrac{2\pi}{3} rh\, dV + \dfrac{\pi}{3} r^2\, dh$.
Partial Differentiation
We sometimes come across quantities that are functions of more than one independent variable.
Partial Differentiation
The little letters here put as subscripts are to show which quantity has been taken as constant in the operation.
Partial Differentiation
But, if you think of it, you will observe that the total variation of $y$ depends on *both* these things at the same time.
Partial Differentiation
This differential equation is of immense importance in mathematical physics.
Partial Differentiation
Another way of indicating that the differentiation has been performed only *partially*, that is, has been performed only with respect to *one* of the independent variables, is to write the differential coefficients with Greek deltas, like $\partial$, instead of little $d$.
Partial Differentiation
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 $x$.
Partial Differentiation
Clearly $x=15$ gives minimum area; $x=10$ gives the maximum, for $\dfrac{d^2 P}{dx^2} = 12x - 150$, which is $+30$ for $x=15$ and $-30$ for $x=10$.
Partial Differentiation
The truck is a rectangular box open at the top. Let $x$ be the length and $y$ be the width; then the depth is $\dfrac{V}{xy}$. The surface area is $S=xy + \dfrac{2V}{x} + \dfrac{2V}{y}$.

Equations

Partial Differentiation
y &= u×v
The simplest concrete case of two variables: y is the product of u and v.
Partial Differentiation
y &= f(u, v)
y depends on two independent variables u and v through a function f.
Partial Differentiation
dy_v &= v\, du
With v held constant, the partial differential of y = uv with respect to u is v du.
Partial Differentiation
dy_u &= u\, dv
With u held constant, the partial differential of y = uv with respect to v is u dv.
Partial Differentiation
\frac{\partial y}{\partial u} &= v
The partial derivative of y = uv with respect to u, treating v as constant, is v.
Partial Differentiation
\frac{\partial y}{\partial v} &= u
The partial derivative of y = uv with respect to v, treating u as constant, is u.
Partial Differentiation
dy_v &= \frac{\partial y}{\partial u}\, du
The partial differential with respect to u, written with the partial derivative.
Partial Differentiation
dy_u &= \frac{\partial y}{\partial v}\, dv
The partial differential with respect to v, written with the partial derivative.
Partial Differentiation
dy = \frac{\partial y}{\partial u}\, du + \dfrac{\partial y}{\partial v}\, dv
When both u and v vary, the total change in y is the sum of its two partial differentials.
Partial Differentiation
dy = \left(\dfrac{dy}{du}\right)\, du + \left(\dfrac{dy}{dv}\right)\, dv
The same total differential written in the alternative notation some books use.
Partial Differentiation
w = 2ax^2 + 3bxy + 4cy^3
Example 1's expression, a polynomial in x and y with constant coefficients a, b, c.
Partial Differentiation
\frac{\partial w}{\partial x} &= 4ax + 3by
Partial derivative of w with respect to x, with y held constant.
Partial Differentiation
\frac{\partial w}{\partial y} &= 3bx + 12cy^2
Partial derivative of w with respect to y, with x held constant.
Partial Differentiation
dw = (4ax+3by)\, dx + (3bx+12cy^2)\, dy
The total differential of w assembled from its two partial derivatives.
Partial Differentiation
z = x^y
Example 2's function: z is x raised to the power y.
Partial Differentiation
\dfrac{\partial z}{\partial x} &= yx^{y-1}
Partial derivative of z = x^y with respect to x, with y held constant.
Partial Differentiation
\dfrac{\partial z}{\partial y} &= x^y × \log_\epsilon x
Partial derivative of z = x^y with respect to y, with x held constant; the result involves the natural logarithm of x.
Partial Differentiation
dz = yx^{y-1}\, dx + x^y \log_\epsilon x \, dy
The total differential of z = x^y.
Partial Differentiation
V=\frac{1}{3} \pi r^2 h
The volume of a cone of base radius r and height h.
Partial Differentiation
\frac{\partial V}{\partial r} &= \dfrac{2\pi}{3} rh
Rate of change of the cone's volume with radius, with height held constant.
Partial Differentiation
\frac{\partial V}{\partial h} &= \dfrac{\pi}{3} r^2
Rate of change of the cone's volume with height, with radius held constant.
Partial Differentiation
dV = \dfrac{2\pi}{3} rh\, dV + \dfrac{\pi}{3} r^2\, dh
As printed, the total differential of the cone's volume; the first term reads dV where dr is evidently intended (erratum flag: the correct term is (2π/3)rh dr).
Partial Differentiation
y &= F(x+at) + f(x-at)
The general form built from two arbitrary functions F and f of x + at and x - at.
Partial Differentiation
\frac{\partial^2 y}{\partial x^2} &= F''(w) + f''(v)
Second partial derivative of y with respect to x, in terms of the second derivatives of F and f.
Partial Differentiation
\frac{\partial^2 y}{\partial t^2} &= F''(w)a^2 + f''(v)a^2
Second partial derivative of y with respect to t, in terms of the second derivatives of F and f.
Partial Differentiation
\frac{\partial^2 y}{\partial t^2} &= a^2\, \frac{\partial^2 y}{\partial x^2}
The second partial derivative in t equals a² times the second partial derivative in x; the book calls this differential equation of immense importance in mathematical physics.
Partial Differentiation
A = \sqrt{s(s-x)(s-y)(s-30+x+y)}
The area of the triangle from its semi-perimeter s and its three sides.
Partial Differentiation
A = \sqrt{15P}
With s = 15 the area reduces to the square root of 15P.
Partial Differentiation
P &= (15-x)(15-y)(x+y-15)
P written as the product of the three side-differences when s = 15.
Partial Differentiation
dP = \dfrac{\partial P}{\partial x}\, dx + \dfrac{\partial P}{\partial y}\, dy
Total differential of P as a function of x and y.
Partial Differentiation
\dfrac{\partial P}{\partial x} = 0 \quad\text{and}\quad \dfrac{\partial P}{\partial y} = 0
For a maximum, both partial derivatives of P must vanish simultaneously.
Partial Differentiation
2xy - 30x + y^2 - 45y + 450 &= 0
The first condition for a stationary point of P, written out.
Partial Differentiation
2xy - 30y + x^2 - 45x + 450 &= 0
The second condition for a stationary point of P, written out.
Partial Differentiation
P = (15-x)^2 (2x-15) = 2x^3 - 75x^2 + 900x - 3375
With x = y, P reduces to the cubic 2x³ - 75x² + 900x - 3375.
Partial Differentiation
6x^2 - 150x + 900 = 0
Setting dP/dx to zero gives the stationary condition for the one-variable cubic.
Partial Differentiation
\dfrac{d^2 P}{dx^2} = 12x - 150
Second derivative of the cubic, used to tell maximum from minimum.
Partial Differentiation
S=xy + \dfrac{2V}{x} + \dfrac{2V}{y}
Surface area of an open rectangular truck of given volume V with length x and width y, depth V/(xy).
Partial Differentiation
dS = \frac{\partial S}{\partial x}\, dx + \frac{\partial S}{\partial y}\, dy
Total differential of the surface area S as a function of x and y.
Partial Differentiation
y - \frac{2V}{x^2} = 0
Stationary condition for S with respect to x, for minimum.
Partial Differentiation
x - \frac{2V}{y^2} = 0
Stationary condition for S with respect to y, for minimum.
Partial Differentiation
S = x^2 + \dfrac{4V}{x}
With x = y, the surface area becomes a function of x alone.
Partial Differentiation
\dfrac{dS}{dx}= 2x - \dfrac{4V}{x^2} =0
Setting dS/dx to zero for a minimum.
Partial Differentiation
x = \sqrt[3]{2V}
The length that minimises the surface area for a given volume V: the cube root of 2V.

Problems

Exercise XV

Exercise XV, problem 10x, p. 181
Find the maximum or minimum of $u = \dfrac{\epsilon^{x+y}}{xy}$.
Printed answer:
- Minimum for $x = y = 1$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes 1
Exercise XV, problem 10y, p. 181
Find the maximum or minimum of $u = \dfrac{\epsilon^{x+y}}{xy}$.
Printed answer:
- Minimum for $x = y = 1$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes 1
Exercise XV, problem 11x, p. 181
Find maximum and minimum of u = y + 2x - 2 _y - _x.
Printed answer:
- Min.: $x = \frac{1}{2}$ and $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes Rational(1,2)
Exercise XV, problem 11y, p. 181
Find maximum and minimum of u = y + 2x - 2 _y - _x.
Printed answer:
- Min.: $x = \frac{1}{2}$ and $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes 2
Exercise XV, problem 121, p. 181
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.
Printed answer:
- Angle at apex $= 90°$; equal sides = length = $\sqrt[3]{2V}$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes, with the problem read into an equation pi/2
Exercise XV, problem 122, p. 181
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.
Printed answer:
- Angle at apex $= 90°$; equal sides = length = $\sqrt[3]{2V}$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes, with the problem read into an equation (2*V)**Rational(1,3)
Exercise XV, problem 1a, p. 180
Differentiate the expression $\dfrac{x^3}{3} - 2x^3y - 2y^2x + \dfrac{y}{3}$ with respect to $x$ alone, and with respect to $y$ alone.
Printed answer:
- $x^3 - 6x^2 y - 2y^2;\quad \frac{1}{3} - 2x^3 - 4xy$.
unverified: no computed check settled this one (yet)
How it was checked
- differentiate: the printed answer does not match the problem x**3 - 6*x**2*y - 2*y**2
Exercise XV, problem 1b, p. 180
Differentiate the expression $\dfrac{x^3}{3} - 2x^3y - 2y^2x + \dfrac{y}{3}$ with respect to $x$ alone, and with respect to $y$ alone.
Printed answer:
- $x^3 - 6x^2 y - 2y^2;\quad \frac{1}{3} - 2x^3 - 4xy$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes Rational(1,3) - 2*x**3 - 4*x*y
Exercise XV, problem 2x, p. 180
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.
Printed answer:
- $2xyz + y^2 z + z^2 y + 2xy^2 z^2$; $2xyz + x^2 z + xz^2 + 2x^2 yz^2$; $2xyz + x^2 y + xy^2 + 2x^2 y^2 z$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 2*x*y*z + y**2*z + z**2*y + 2*x*y**2*z**2
Exercise XV, problem 2y, p. 180
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.
Printed answer:
- $2xyz + y^2 z + z^2 y + 2xy^2 z^2$; $2xyz + x^2 z + xz^2 + 2x^2 yz^2$; $2xyz + x^2 y + xy^2 + 2x^2 y^2 z$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 2*x*y*z + x**2*z + x*z**2 + 2*x**2*y*z**2
Exercise XV, problem 2z, p. 180
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.
Printed answer:
- $2xyz + y^2 z + z^2 y + 2xy^2 z^2$; $2xyz + x^2 z + xz^2 + 2x^2 yz^2$; $2xyz + x^2 y + xy^2 + 2x^2 y^2 z$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 2*x*y*z + x**2*y + x*y**2 + 2*x**2*y**2*z
Exercise XV, problem 3a, p. 180
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 $\dfrac{\partial^2r}{\partial x^2} + \dfrac{\partial^2r}{\partial y^2} + \dfrac{\partial^2r}{\partial z^2}$.
Printed answer:
- $\dfrac{1}{r} \{ \left(x - a\right) + \left( y - b \right) + \left( z - c \right) \} = \dfrac{ \left( x + y + z \right) - \left( a + b + c \right) }{r}$; $\dfrac{3}{r}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes, with the problem read into an equation ((x-a) + (y-b) + (z-c))/sqrt((x-a)**2 + (y-b)**2 + (z-c)**2)
Exercise XV, problem 3b, p. 180
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 $\dfrac{\partial^2r}{\partial x^2} + \dfrac{\partial^2r}{\partial y^2} + \dfrac{\partial^2r}{\partial z^2}$.
Printed answer:
- $\dfrac{1}{r} \{ \left(x - a\right) + \left( y - b \right) + \left( z - c \right) \} = \dfrac{ \left( x + y + z \right) - \left( a + b + c \right) }{r}$; $\dfrac{3}{r}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 3/sqrt((x-a)**2 + (y-b)**2 + (z-c)**2)
Exercise XV, problem 4, p. 180
Find the total differential of $y=u^v$.
Printed answer:
- $dy = vu^{v-1}\, du + u^v \log_\epsilon u\, dv$.
unverified: no computed check settled this one (yet)
How it was checked
- other: the check does not fit this problem v*u**(v-1)*du + u**v*log(u)*dv
Exercise XV, problem 51, p. 181
Find the total differential of $y=u^3 \sin v$; of $y = (\sin x)^u$; and of $y = \dfrac{\log_\epsilon u}{v}$.
Printed answer:
- $dy = 3\sin v u^2\, du + u^3 \cos v\, dv$, $dy = u \sin x^{u-1} \cos x\, dx + (\sin x)^u \log_\epsilon \sin x du$, $dy = \dfrac{1}{v}\, \dfrac{1}{u}\, du - \log_\epsilon u \dfrac{1}{v^2}\, dv$.
unverified: no computed check settled this one (yet)
How it was checked
- other: the check does not fit this problem 3*sin(v)*u**2*du + u**3*cos(v)*dv
Exercise XV, problem 52, p. 181
Find the total differential of $y=u^3 \sin v$; of $y = (\sin x)^u$; and of $y = \dfrac{\log_\epsilon u}{v}$.
Printed answer:
- $dy = 3\sin v u^2\, du + u^3 \cos v\, dv$, $dy = u \sin x^{u-1} \cos x\, dx + (\sin x)^u \log_\epsilon \sin x du$, $dy = \dfrac{1}{v}\, \dfrac{1}{u}\, du - \log_\epsilon u \dfrac{1}{v^2}\, dv$.
unverified: no computed check settled this one (yet)
How it was checked
- other: the check does not fit this problem u*sin(x)**(u-1)*cos(x)*dx + sin(x)**u*log(sin(x))*du
Exercise XV, problem 53, p. 181
Find the total differential of $y=u^3 \sin v$; of $y = (\sin x)^u$; and of $y = \dfrac{\log_\epsilon u}{v}$.
Printed answer:
- $dy = 3\sin v u^2\, du + u^3 \cos v\, dv$, $dy = u \sin x^{u-1} \cos x\, dx + (\sin x)^u \log_\epsilon \sin x du$, $dy = \dfrac{1}{v}\, \dfrac{1}{u}\, du - \log_\epsilon u \dfrac{1}{v^2}\, dv$.
unverified: no computed check settled this one (yet)
How it was checked
- other: the check does not fit this problem (1/v)*(1/u)*du - log(u)*(1/v**2)*dv
Exercise XV, problem 6, p. 181
Verify that the sum of three quantities $x$, $y$, $z$, whose product is a constant $k$, is maximum when these three quantities are equal.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check
Exercise XV, problem 7x, p. 181
Find the maximum or minimum of the function u = x + 2xy + y.
Printed answer:
- Minimum for $x = y = -\frac{1}{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: the printed answer does not match the problem -Rational(1,2)
Exercise XV, problem 7y, p. 181
Find the maximum or minimum of the function u = x + 2xy + y.
Printed answer:
- Minimum for $x = y = -\frac{1}{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: the printed answer does not match the problem -Rational(1,2)
Exercise XV, problem 8a, p. 181
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.
Printed answer:
- (*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. = $2.54$.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes, with the problem read into an equation 1
- evaluate: passes 1
- evaluate: passes 1
Exercise XV, problem 8b, p. 181
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.
Printed answer:
- (*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. = $2.54$.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: passes, with the problem read into an equation 2/pi
- evaluate: the printed answer does not match the problem 2/pi
- evaluate: passes 2/pi
- evaluate: PASS-LOOSE 2/pi
Exercise XV, problem 9x, p. 181
Divide $\pi$ into $3$ parts such that the continued product of their sines may be a maximum or minimum.
Printed answer:
- All three parts equal; the product is maximum.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: the printed answer does not match the problem pi/3
Exercise XV, problem 9y, p. 181
Divide $\pi$ into $3$ parts such that the continued product of their sines may be a maximum or minimum.
Printed answer:
- All three parts equal; the product is maximum.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: the printed answer does not match the problem pi/3