Linear Approximation of 3 Variables: Formula Check

[tandoorichicken]

My book gives a formula for linear approximation of two independent variables, but I needed one for three. So I modified the formula given in the book, but I need someone to please just quickly see if it looks okay.

Given:
$$f(x,y)=z=f(x_0,y_0)+(\frac{\partial f}{\partial x} (x_0,y_0)) (x-x_0) + (\frac{\partial f}{\partial y} (x_0,y_0)) (y-y_0)$$

Modified:
$$f(x,y,z)=f(x_0,y_0,z_0)+(\frac{\partial f}{\partial x} (x_0,y_0,z_0)) (x-x_0) + (\frac{\partial f}{\partial y} (x_0,y_0,z_0)) (y-y_0)+(\frac{\partial f}{\partial z} (x_0,y_0,z_0)) (z-z_0)$$

Does this look alright?
It looks fine to me but I'm prone to overlooking glaring errors

 

[benorin]

Beautiful.

 

[tandoorichicken]

thanks a lot.

 

[Hurkyl]

When you have time, you might want to see if you can rigorously justify that expression!