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General three-variable relationship In the implicit
Chapter 11, Problem 63AE(choose chapter or problem)
General three-variable relationship In the implicit relationship F(x, y, z) = 0, any two of the variables may be considered independent, which then determines the third variable. To avoid confusion, we may use a subscript to indicate which variable is held fixed in a derivative calculation, for example, \(\left(\frac{\partial z}{\partial x}\right)_{y}\) means that y is held fixed in taking the partial derivative of z with respect to x. (In this context, the subscript does not mean a derivative.)
a. Differentiate F(x, y, z) = 0 with respect to x holding y fixed to show that \(\left(\frac{\partial z}{\partial x}\right)_{y}=-\frac{F_{x}}{F_{z}} .\)
b. As in part (a), find \(\left(\frac{\partial y}{\partial z}\right)_{x} \text { and }\left(\frac{\partial x}{\partial y}\right))
c. Show that \(\left(\frac{\partial z}{\partial x}\right)_{y}\left(\frac{\partial y}{\partial z}\right)_{x}\left(\frac{\partial x}{\partial y}\right)_{z}=-1 .\).
d. Find the relationship analogous to part (c) for the case F(w, x, y, z) = 0.
Questions & Answers
QUESTION:
General three-variable relationship In the implicit relationship F(x, y, z) = 0, any two of the variables may be considered independent, which then determines the third variable. To avoid confusion, we may use a subscript to indicate which variable is held fixed in a derivative calculation, for example, \(\left(\frac{\partial z}{\partial x}\right)_{y}\) means that y is held fixed in taking the partial derivative of z with respect to x. (In this context, the subscript does not mean a derivative.)
a. Differentiate F(x, y, z) = 0 with respect to x holding y fixed to show that \(\left(\frac{\partial z}{\partial x}\right)_{y}=-\frac{F_{x}}{F_{z}} .\)
b. As in part (a), find \(\left(\frac{\partial y}{\partial z}\right)_{x} \text { and }\left(\frac{\partial x}{\partial y}\right))
c. Show that \(\left(\frac{\partial z}{\partial x}\right)_{y}\left(\frac{\partial y}{\partial z}\right)_{x}\left(\frac{\partial x}{\partial y}\right)_{z}=-1 .\).
d. Find the relationship analogous to part (c) for the case F(w, x, y, z) = 0.
ANSWER:Solution 63AE
Step 1 of 4:
- In this problem we need to differentiate f(x,y,z) = 0 with respect to x holding y fixed and we have to show that .
Given y is fixed , so F(x , y, z) = 0 can be written as F( x, z(x)) = 0.Since x is an independent variable.
To find , differentiate both sides of F(x , z(x)) = 0 with respect to x.
For this F(x , z(x)) = 0 .
, since .