Hello,Can someone please show how to solve this problem?Thank you Problem 5.4 Partial derivativesa) Calculate the first-order partial derivatives (fa, fy, and fz) of:2/3= (x³ + ( ² )) ²/³,andf(x, y, z) = (x³ -f(x, y, z)= ey In (y).b) Calculate the second-order partial derivatives (fax, fyy, fzz) and show that fry = fyx ofthe functions in (a).c) The heat equation (also known as the diffusion equation), expressed in one spatial di-mension x and time t, readsaf 8² f?х2'Ət=It is relevant for the description of many physical processes, such as how temperature(or the concentration of a chemical) is distributed in space and time. Show that the twofunctions, f and g, obey the heat equation:f(x, t)g(x, t) =et sin(x) + x.e-x²/4t√4t

fx,y,z=x3+y223fx=233x2x3+y2-13 =2x2x3+y213fy=12·23x3+y2-13…

Answered: a) Calculate the first-order partial…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Hello,

Can someone please show how to solve this problem?

Thank you

Problem 5.4 Partial derivatives
a) Calculate the first-order partial derivatives (fa, fy, and fz) of:
2/3
= (x³ + ( ² )) ²/³,
and
f(x, y, z) = (x³ -
f(x, y, z)= ey In (y).
b) Calculate the second-order partial derivatives (fax, fyy, fzz) and show that fry = fyx of
the functions in (a).
c) The heat equation (also known as the diffusion equation), expressed in one spatial di-
mension x and time t, reads
af 8² f
?х2'
Ət
=
It is relevant for the description of many physical processes, such as how temperature
(or the concentration of a chemical) is distributed in space and time. Show that the two
functions, f and g, obey the heat equation:
f(x, t)
g(x, t) =
et sin(x) + x.
e-x²/4t
√4t

Expert Solution

Step 1: Calculating the first order derivative

$f (x, y, z) = {(x^{3} + \frac{y}{2})}^{\frac{2}{3}} f_{x} = \frac{2}{3} (3 x^{2}) {(x^{3} + \frac{y}{2})}^{- \frac{1}{3}} = \frac{2 x^{2}}{{(x^{3} + \frac{y}{2})}^{\frac{1}{3}}} f_{y} = \frac{1}{2} \cdot \frac{2}{3} {(x^{3} + \frac{y}{2})}^{- \frac{1}{3}} = \frac{1}{3 {(x^{3} + \frac{y}{2})}^{\frac{1}{3}}} f_{z} = 0 f (x, y, z) = e^{x y} \ln (y) f_{x} = y e^{x y} \ln (y) f_{y} = x e^{x y} \ln (y) + \frac{e^{x y}}{y} f_{z} = 0$