Given F(x,y,z) = x sin(zy) - e^{x+z} = 0. Show this is graph of x=g(y,z) near (0,1,0)

[sunderwood2]

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1. Let a surface be defined by F (x, y, z) = x sin(yz) - e ^x+z = 0.

(a) Show that this surface is the graph of a function x = g (y, z) near the point (0, 1, 0).

(b) Find \(\displaystyle \, \dfrac{\partial g}{\partial y}\,\) and \(\displaystyle \, \dfrac{\partial g}{\partial z}\, \) at (0, 1, 0).

 

[Deleted member 4993]

1. Let a surface be defined by F (x, y, z) = x sin(yz) - e ^x+z = 0.

(a) Show that this surface is the graph of a function x = g (y, z) near the point (0, 1, 0).

(b) Find \(\displaystyle \, \dfrac{\partial g}{\partial y}\,\) and \(\displaystyle \, \dfrac{\partial g}{\partial z}\, \) at (0, 1, 0).

These look like test questions!!

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[HallsofIvy]

The first problem asks you to show that it is possible to solve the given equation for x, as a function of y and z, at least for some region around the given point. Under what conditions is that possible?

(For example, do you see that it is possible to solve \(\displaystyle y= x^2\) for x, as a function of y, in a small region around every point except (0, 0)? What is special about that point?)