- #1

Buri

- 273

- 0

f'(a;u) = lim [t -> 0] (1/t)[f(a + tu) - f(a)]

And the partial derivative to be defined as the directional derivative in the direction of u = e_i.

My text, Analysis on Manifolds by Munkres, says that the directional derivative exists if the limit exists which I understand to mean that the right and left handed limits exists and are equal to each other. However, I asked here earlier about f(x,y) = |x| + |y| and was told the directional derivatives definition is t+ -> 0, so in fact the directional derivatives of f(x,y) do exist! But with even tweaking my definition to t+ then my partial derivatives get messed up because apparently I have to check those for t+ and t- and must equal each other.

So I'm really really confused. Can someone help me out on this? I would think that Munkres would define them correctly, so I'm probably not understanding something. Any help?