For a function f:U→R (U⊆Rn some open set) and a point x=(x1,...,xn), define
is a function of one variable
when t is near xi
, and we can do calculus
with it. The partial derivative
of f with respect to xi
is defined to be
∂f/∂xi = dfi,x/dt.
Note the unfortunate double usage of x and especially xi to denote both a vector or a scalar variable and a particular value of that variable. Unfortunately, this practice is so common that I feel I must abide by it.
This is a function ∂f/∂xi
If the derivative on the RHS
does not exist, the partial derivative on the LHS
By tradition, the same letters are re-used, instead of new ones like t. Your multivariable calculus textbook may well follow this "convention", and give a seemingly different definition. The only purpose of this is to make a confusing subject more confusing.
Another notation for ∂f/∂xi is fxi. This notation is particularly useful for taking multiple partial derivatives: for a function f(x,y),
fxx = ∂2f/∂x2 =
fxy = ∂2f/∂y∂x =
fyx = ∂2f/∂x∂y =
fyy = ∂2f/∂y2 =
In particular, the Laplacian
of f is Δf =
WHEN f is differentiable, it turns out that
∇f = (∂f/∂x1,...,∂f/∂xn).
But even if all partial derivatives exist
, f might not be differentiable there -- be careful
Contrary to what you might expect (or even read about in various places, once upon a time even on E2...), it is not necessarily true that ∂2f/∂xi∂xj = ∂2f/∂xj∂xi. You cannot, in general, interchange the order of the derivatives.