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## Functions of 2 variables.

Definition 6.9   Let f be a function of two real variables, defined on a neighborhhod of the point P0 (x0,y0). Suppopse that f has partial derivatives of first order at P0. The vector is called the gradient of f at P0.

Example 6.10   Let and . Then and . The gradient of f at P0 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 P0 in the direction of is the number:

Example 6.12   Let f(x,y)=4- x2-y2, P0(1,1). The graph of f is displayed on Figure  4(a).

We have: fx=-2x and fy=-2y. At the point P0, 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 z-axis is the same.

Example 6.13   Let f(x,y)=4- x2-2y2, P0(1,1). The graph of f is displayed on Figure  4(a).

We have: fx=-2x and fy=-4y. At the point P0, the gradient of f is .

• In the direction of : .
• In the direction of : .
Compare this result to the previous example.

Proposition 6.14   Let f, P0, and be as above (see Def  6.11). We denote by the angle of the vectors and . Then:
1.
The directional derivative has its greatest value when , i.e. when the direction of and the direction of the gradient are identical. This means that f increases most fastly at a point P0 in the direction of the gradient.
2.
The directional derivative has its least value when , i.e. when the direction of and the direction of the gradient are opposite. This means that f decreases most fastly at a point P0 in the direction opposite to the gradient's direction.
3.
If is orthogonal to the gradient , then the directional derivative is equal to 0.

Example 6.15   Let f(x,y)=x2-y2. Then: fx=2x and fy=-2y. At the point P0(1,0), we have: . Thus at P0, the greatest value of the directional derivative is afforded in the direction of , i.e. in the positive direction of the x-axis (it points to the right in Figure  5).

Proposition 6.16   At every point P0 in the domain of f, the vector is normal to the level curve through P0.

Example 6.17   Let f(x,y)=x2+y2 and P0(1,2). Then and . This vector is normal to the level curve through P0, whose equation is x2+y2=5. The line Through P0 and normal to is the tangent to the circle at P0; its equation is 2(x-1)+4(y-2)=0, i.e. x+2y-3=0.

Next: Functions of 3 variables. Up: Directional derivative and Gradient Previous: Directional derivative and Gradient
Noah Dana-Picard
2001-05-30
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