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a) Calculate the first-order partial derivatives (fa, fy, and fz) of: 2/3 f (x, y, z) = (x³ + (²)) ²/³, and f(x, y, z)= ey In (y). b) Calculate the second-order partial derivatives (fax, fyy, fzz) and show that fry 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 Ət 8² f əx²* - fyz of et sin(x) + x. e-z²/4t √4t = 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) =

a) Calculate the first-order partial derivatives (fa, fy, and fz) of: 2/3 f (x, y, z) = (x³ + (²)) ²/³, and f(x, y, z)= ey In (y). b) Calculate the second-order partial derivatives (fax, fyy, fzz) and show that fry 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 Ət 8² f əx²* - fyz of et sin(x) + x. e-z²/4t √4t = 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) =

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.CR: Chapter 9 Review
Problem 4CR
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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
Transcribed Image Text: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
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