General three-variable relationship In the implicit

Chapter 11, Problem 63AE

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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.

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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:

  1. 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 .

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