Michael Corral: Vector Calculus

80 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES
Definition 2.6. For a real-valued function f(x,y), the gradient of f, denoted by∇f,
is the vector ( ∂f
)
∇f=
∂x ,∂f
(2.12)
∂y
in 2 . For a real-valued function f(x,y,z), the gradient is the vector
∇f=
in 3 . The symbol∇is pronounced “del”. 5
Corollary 2.3. D v f= v·∇f
( ∂f
)
∂x ,∂f ∂y ,∂f ∂z
(2.13)
Example 2.15. Find
(
the directional derivative of f(x,y)= xy 2 +x 3 y at the point (1,2) in
1
the direction of v= √, √2
1
). 2
Solution: We see that∇f= (y 2 +3x 2 y,2xy+ x 3 ), so
( )
1
D v f(1,2)=v·∇f(1,2)= √
1
, √2 ·(2 2 +3(1) 2 (2),2(1)(2)+1 3 )= √ 15
2 2
A real-valued function z= f(x,y) whose partial derivatives ∂f ∂f
∂x
and
∂y
exist and are
continuous is called continuously differentiable. Assume that f(x,y) is such a function
and that∇f 0. Let c be a real number in the range of f and let v be a unit vector in
2 which is tangent to the level curve f(x,y)=c(see Figure 2.4.1).
y
v
∇f
f(x,y)=c
0
x
Figure 2.4.1
5 Sometimes the notation grad(f) is used instead of∇f.