Question

Def. 1 : A function f of two variables is said to be differentiable at (x0,y0) provided fx(x0,y0) both exist and
[insert equation 1 here]

Def 2: A function f of three variables is said to be differentiable at (x0,y0,z0) provided fx(x0,y0,z0), fy(x0,y0,z0) and fz(x0,y0,z0) exist and
[insert eq 2 here].

Use the above two def (s) to prove
a) a constant function of 2 or 3 variables is differentiable everywhere.
b) a linear function of 2 or 3 variables is differentiable everywhere.

Answer 1

eq1:

\[\lim_{(\Delta x,\Delta y) \rightarrow (0,0)} \frac{ \Delta f - f _{x} (x _{0}, y _{0}) \Delta x - f _{y}(x _{0}, y _{0}) \Delta y}{ \sqrt(\Delta x)^2 + (\Delta y)^2 } =0\]

Answer 2

the second eq is very similar, except where it was xy, it is xyz.

Answer 3

I think I know how to prove this, I'm just not quite sure what it means by a constant function of 2 or 3 variables.

Answer 4

Constant function? D: Hmm do they mean like,\[\large f(x,y)=c\]

Answer 5

sorry, my computer crashed. :(

Answer 6

maybe, thanks, I'll give it a go.

Answer 7

yup, definitely. Thanks again.