Suppose that the real variables p, v, t, and u satisfy the equations f(p, v, t, u) = 0, g(p, v, t, u) = 0, and that these two equations can be solved for any two of the four variables as functions of the other two. Then the symbol du/dp, for example, is ambiguous. We denote by (du/Op)t the partial derivative of u with respect to p, with u and v considered as functions of p and t, and by (du/dp), the partial derivative of u with respect to p, with u and t considered as functions of p and v. With this notation, show that ди ди at du dp It All functions that appear in this problem may be assumed to be defined and differen- tiable in a suitable set to make the arguments work.
Suppose that the real variables p, v, t, and u satisfy the equations f(p, v, t, u) = 0, g(p, v, t, u) = 0, and that these two equations can be solved for any two of the four variables as functions of the other two. Then the symbol du/dp, for example, is ambiguous. We denote by (du/Op)t the partial derivative of u with respect to p, with u and v considered as functions of p and t, and by (du/dp), the partial derivative of u with respect to p, with u and t considered as functions of p and v. With this notation, show that ди ди at du dp It All functions that appear in this problem may be assumed to be defined and differen- tiable in a suitable set to make the arguments work.
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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Transcribed Image Text:Suppose that the real variables p, v, t, and u satisfy the equations
f(p, v, t, u) = 0,
g(p, v,t, u) = 0,
and that these two equations can be solved for any two of the four variables as functions
of the other two. Then the symbol du/dp, for example, is ambiguous. We denote
by (du/dp)t the partial derivative of u with respect to p, with u and v considered as
functions of p and t, and by (8u/@p), the partial derivative of u with respect to p, with
u and t considered as functions of p and v. With this notation, show that
).- G).C).- G),C). - O).
ди
dt
dt
dt
dp
dp
All functions that appear in this problem may be assumed to be defined and differen-
tiable in a suitable set to make the arguments work.
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