Taylor's theorem for function F(x,y,z)

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This is related to a book that I am reading (Calculus of Variations).

There is a function $F(x,y,z)$ which arguments are differentiable. Now, how I can derive a Taylor's polynomial for function $F(x,y,z)$?

I am familiar with the simple version of Taylor's theorem with one variable, but not multivariable version.

I have hard time to understand what the degree means in Taylor's theorem. I have seen definitions for second, third and higher degree Taylor theorems, but it is still bit odd for me.

I would be very pleased old man if someone could open this a bit for me.