7Directional derivative

IA Differential Equations



7.3 Taylor series for multi-variable functions
Suppose we have a function
f
(
x, y
) and a point x
0
. Now consider a finite
displacement δs along a straight line in the x, y plane. Then
δs
d
ds
= δs ·
The Taylor series along the line is
f(s) = f(s
0
+ δs)
= f(s
0
) + δs
df
ds
+
1
2
(δs)
2
d
2
f
ds
2
= f(s
0
) + δs · f +
1
2
δs
2
(ˆs ·)(ˆs · )f.
We get
δs · f = (δx)
f
x
+ (δy)
f
y
= (x x
0
)
f
x
+ (y y
0
)
f
y
and
δs
2
(ˆs · )(ˆs · )f = (δs · )(δs · )f
=
δx
x
+ δy
y
δx
f
x
+ δy
f
y
= δx
2
2
f
x
2
+ 2δxδy
2
f
x∂y
+ δy
2
2
f
y
2
=
δx δy
f
xx
f
xy
f
yx
f
yy
δx
δy
Definition (Hessian matrix). The Hessian matrix is the matrix
∇∇f =
f
xx
f
xy
f
yx
f
yy
In conclusion, we have
f(x, y) = f(x
0
, y
0
) + (x x
0
)f
x
+ (y y
0
)f
y
+
1
2
[(x x
0
)
2
f
xx
+ 2(x x
0
)(y y
0
)f
xy
+ (y y
0
)
2
f
yy
]
In general, the coordinate-free form is
f(x) = f(x
0
) + δx · f(x
0
) +
1
2
δx · ∇∇f · δx
where the dot in the second term represents a matrix product. Alternatively, in
terms of the gradient operator (and real dot products), we have
f(x) = f(x
0
) + δx · f(x
0
) +
1
2
[(f · δx)] · δx