Chapter 15, Partial Differentiation Video Solutions, Engineering Mathematics Through Applications

Problem 1

Find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for each of the following functions:
a $f(x, y)=x^{2}+y^{2}$
b $f(x, y)=\sin (x)+\cos (y)$
c $f(x, y)=x^{3}+y^{3}+3 x y$
d $f(x, y)=2 y-\frac{1000}{x}$

Will Erickson

Numerade Educator

04:04

Problem 2

[materials] A beam is subject to a uniform load of $w$ per unit length and a concentrated load $P$. The bending moment $M$ at a distance $x$ from one end is given by
$$
M=P x+\frac{w x^{3}}{3}
$$
Determine $\frac{\partial M}{\partial P}$ and $\frac{\partial M}{\partial x}$.

Angela Guo

Numerade Educator

07:55

Problem 3

[materials] Castigliano's theorem says the displacement, $\Delta$, under a force $P$ is given by
$$
\Delta=\frac{\partial U}{\partial P}
$$
where $U$ is the internal strain energy of the body. For the following, find $\Delta$ :
a $U=\frac{P^{2} L}{2 A E}$
b $U=\frac{9 P^{2} L^{3}}{96 E I}$

Sriram Soundarrajan

Numerade Educator

01:20

Problem 4

(structures] Given the stress function, $\Omega(x, y)$, as (A and B are constants)
$$
\Omega(x, y)=A x^{2} y^{4}+B x^{4} y^{2}
$$
find the stresses, $\sigma_{x}$ and $\sigma_{y}$, where
$$
\sigma_{x}=\frac{\partial^{2} \Omega}{\partial y^{2}} \text { and } \sigma_{y}=\frac{\partial^{2} \Omega}{\partial x^{2}}
$$

Hast Aggarwal

Numerade Educator

03:55

Problem 5

[fluid mechanics] The continuity equation (partial differential equation) is defined as
$$
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0
$$
where $u, v$ are velocities of the fluid in the $x, y$ directions respectively. Show that each of the following satisfy the continuity equation:
a $u=y^{2}-x^{2}, \quad v=2 x y$
b $u=\tan ^{-1}\left(\frac{y}{x}\right), \quad v=\frac{1}{2} \ln \left(x^{2}+y^{2}\right)$
c $u=\frac{-2 y}{(1+x)^{2}+y^{2}}, v=\frac{1-x^{2}-y^{2}}{(1+x)^{2}+y^{2}}$

Sriram Soundarrajan

Numerade Educator

12:05

Problem 6

[fluid mechanics] The stream function, $\psi(x, y)$, is related to the velocity components $u$ and $v$ of the fluid flow by
$u=\frac{\partial \psi}{\partial y}$ and $v=-\frac{\partial \psi}{\partial x}$
If $\psi=\frac{1}{2} \ln \left(x^{2}+y^{2}\right)$, find $u$ and $v$
The flow is irrotational if $\psi$ satisfies Laplace's equation which is given by
$$
\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}=0
$$
Show that the flow is irrotational for
$$
\psi=\frac{1}{2} \ln \left(x^{2}+y^{2}\right)
$$
Wh Questions 7 to 9 are in the field of [thermodynamics].
( $P$ is pressure, $V$ is volume, $T$ is temperature and $R$ is the gas constant)

Sheh Lit Chang

University of Washington

01:56

Problem 7

The following terms are used in thermodynamics:
Isothermal compressibility $\kappa=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)$
Thermal expansion coefficient
$$
\beta=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)
$$
Show that for the ideal gas equation, $P V=R T$, we have
$\kappa=\frac{1}{P}$ and $\beta=\frac{1}{T}$

Dan Ni

Numerade Educator

02:06

Problem 8

An equation involving specific heats $c_{\mathrm{p}}$, $c_{\mathrm{v}}$ is defined as
$$
c_{\mathrm{p}}-c_{\mathrm{v}}=T\left(\frac{\partial V}{\partial T}\right)\left(\frac{\partial P}{\partial T}\right)
$$
For the equation
$$
\frac{R T}{P}=V+\frac{K}{R T}
$$
( $K$ is a constant)
find $c_{\mathrm{p}}-c_{\mathrm{v}}$.

Aman Gupta

Numerade Educator

04:56

Problem 9

Van der Waals' equation is given by
$$
P=\frac{R T}{V-b}-\frac{a}{V^{2}}
$$
where $a$ and $b$ are constants. The critical isotherm occurs when $\frac{\partial P}{\partial V}=0$ and $\frac{\partial^{2} P}{\partial V^{2}}=0$
A point which satisfies these two equations is called a 'critical point'. At the critical point the volume and temperature are $V_{\mathrm{c}}$ and $T_{\mathrm{c}}$ respectlvely. Find $a$ and $b$ in their simplest form for a critical point.

Farhana Sharmin

Numerade Educator

01:26

Problem 10

Y [vibrations] The displacement, $u(x, t)$, of a rod is a function of position $x$ and time $t$ :
$$
\begin{aligned}
u(x, t)=&\left(A \sin \left(\frac{\omega}{\alpha} x\right)+B \cos \left(\frac{\omega}{\alpha} x\right)\right) \\
& \times[\operatorname{Csin}(\omega t)+D \cos (\omega t)]
\end{aligned}
$$
where $\alpha=\sqrt{\frac{E}{\rho}},(E$ is modulus of elasticity, $\rho$ is density of rod and $\omega$ is natural frequency of vibration). Show that $u(x, t)$ satisfies the wave equation:
$$
\alpha^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}
$$

Manik Pulyani

Numerade Educator

01:45

Problem 11

Find $\left(\frac{\partial^{2} f}{\partial x^{2}}\right)\left(\frac{\partial^{2} f}{\partial y^{2}}\right)-\left(\frac{\partial^{2} f}{\partial x \partial y}\right)^{2}$ for the following functions:
a $f(x, y)=x^{3}+y^{3}-3 x y$
b $f(x, y)=2 x y+\frac{2000}{x}+\frac{2000}{y}$

Zachary Mitchell

Numerade Educator