| ↔ previous | next ≈ |
Definition (directional derivative)
Let f : D → R, D ⊂ Rm, A = (a1,a2,..., am) ∈ IntD and let v = (v1,v2,..., vm) be a non zero vector. The directional derivative of f in the direction of vector v is the limit (if it exists and is finite)
denoted by
or fv(A).
Example
Let 
Then
Corollary 1
The partial derivatives of the first order are a specific case of
directional derivatives. Precisely: if ek=(0,...,0,1,0,...,0)
(1 is on the k-th position) is the k-th unit
basis vector Rm, then 
.
Because
Corollary 2
The existence of all directional derivatives f at A does not guarantee the continuity of f at the point A.
Example
Let 
.
The function f is not continuous at (0,0),
because
(see the definition of a continuous function).
But for any vector v=(v1,v2) ≠ (0,0) the derivative 
exists,
The gradient of a function allows us to easily find the functions
directional derivatives.
Theorem (an expression to find directional derivatives)
Let D ⊂ Rm and
let f : D ⎯→
R be a function which has the following partial
derivatives 
in some neighbourhood of the point A, and which
are continuous at A. Then 
for all non zero vectors v ∈
Rm.
Example
• Find 
for 
, when A=(0,1,3),
v=(-1,1,2).
We know that że
(the first example following the definition of the
gradient). Thus 
.
• Let 
. We find .
Then 
, thus 
. So
.
| ↔ previous | next ≈ |