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Definition 6.9
Let
f be a function of two real variables, defined on a neighborhhod of the point
P_{0} (
x_{0},
y_{0}). Suppopse that
f has partial derivatives of first order at
P_{0}. The vector
is called
the gradient of
f at
P_{0}.
Example 6.10
Let
and
.
Then
and
.
The gradient of
f at
P_{0} is the vector
.
Definition 6.11
We use the settings of Def
6.9;
let
be a unit vector.
The
directional derivative of
f at
P_{0} in the direction of
is the number:
Example 6.12
Let
f(
x,
y)=4
x^{2}
y^{2},
P_{0}(1,1). The graph of
f is displayed on Figure
4(a).
We have: f_{x}=2x and f_{y}=2y. At the point P_{0}, the gradient of f is
.
 In the direction of
:
.
 In the direction of
:
.
This result is not surprising as the graph of
f is a paraboloid of revolution, i.e. the behaviour of
f in any direction perpendicular to the
zaxis is the same.
Figure 4:
Paraboloids.

Example 6.13
Let
f(
x,
y)=4
x^{2}2
y^{2},
P_{0}(1,1). The graph of
f is displayed on Figure
4(a).
We have: f_{x}=2x and f_{y}=4y. At the point P_{0}, the gradient of f is
.
 In the direction of
:
.
 In the direction of
:
.
Compare this result to the previous example.
Example 6.15
Let
f(
x,
y)=
x^{2}
y^{2}. Then:
f_{x}=2
x and
f_{y}=2
y. At the point
P_{0}(1,0), we have:
.
Thus at
P_{0}, the greatest value of the
directional derivative is afforded in the direction of
,
i.e. in the positive direction of the
xaxis (it points to the right in Figure
5).
Figure 5:
Two special directions.

Proposition 6.16
At every point
P_{0} in the domain of
f, the vector
is normal to the level curve through
P_{0}.
Example 6.17
Let
f(
x,
y)=
x^{2}+
y^{2} and
P_{0}(1,2). Then
and
.
This vector is normal to the level curve
through
P_{0}, whose equation is
x^{2}+
y^{2}=5. The line Through
P_{0} and normal to
is the tangent to the circle
at
P_{0}; its equation is
2(
x1)+4(
y2)=0, i.e.
x+2
y3=0.
Figure 6:
A tangent to a level curve.

Next: Functions of 3 variables.
Up: Directional derivative and Gradient
Previous: Directional derivative and Gradient
Noah DanaPicard
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