Calculus III:

Vector functions
Vector fields
Gradient vector field
Conservative vector fields

1. Definition

A vector field is a function vector that assigns to each point M in the space a vector (M).

In two-dimension space,
M(x,y): (M) = (x,y).

In three-dimension space,
if M(x,y,z):(M) = (x,y,z).

The standard notation of a vector field (M)
is built as :

• In two-dimention space:

(M) = P(x,y) + Q(x,y,z)

• In three-dimention space:

(M) = P(x,y,z) + Q(x,y,z) + R(x,y,z)

2. Example

Let's sketch vector field (M) = - (x/2) + (y/2)

To graph the vector field, let's do evaluations.

• (0, 0) =

• (1, - 1) = - (1/2) - (1/2)

At the point (1, - 1), we plot the vector
- (1/2) - (1/2).

• (- 4, - 2) = 2 - 1

At the point (- 4, - 2), we plot the vector
2 - .

(1, 2) = - (1/2) + 1.

At the point (1, 2), we plot the vector
- (1/2) + .



We can continue in this fashion plotting vectors for several points and we’ll get an almost complete sketch of the vector field.

3. The special case: The gradient vector

The special case of vector field function is the the gradient vector.

Given a function f(x, y, z), the gradient vector is written ∇f , and defined by:

∇f = f_x + f_y + f_z

Where

f_x = ∂f/∂x, f_y = ∂f/∂y, and f_z = ∂f/∂z,

Notice that f(x, y, z) is called scalar function . For a two-variable function f(x,y), we just need to drop off the third component of the vector.

4. Conservative vector fields

A vector field is called a conservative vector field if there exists a function ƒ such that = ∇ƒ.

If is a conservative vector field then the function, Æ’ is called a potential function for .

That is, a vector field is conservative if it is a gradient vector field for some function.

Exemple

The vector field = y + x is a conservative vector field with a potential function of Æ’(x, y) = xy.
Indeed ∇ƒ = 〈 y, x 〉.