Okay guys and gals, now that you have some definitions of vector fields, it's time that we meet the famous del operator
. In doing so, we will explore the vector-equivalent derivative in 3-dimenstional space.Time we move from the number line to 3D space.
VECTOR INTEGRAL CALCULUS >
THE GRADIENT VECTOR FIELD.2
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THE GRADIENT VECTOR FIELD.1
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CONCEPT
Let
 be a real-value function of three variables. In the context of vectors, this function is called a scalar field. The gradient of  , denoted by grad  or , is a vector field extracted from  according to the operation,
where each of these partial derivatives are defined. The notation It also works on functions of two variable where
Ensure that you are taking partial derivatives which basically means holding the other variable fixed and differentiating accordingly to the w.r.t term, i.e.,
Let us look at a simple example. Suppose
then
I will now define a new term which you guys may be initially confused at. Don’t worry, you will be more familiar with it as we proceed in our learning. The gradient of In single variable calculus, the derivative of
Now, let’s us move into 3-dimensional space and think of the vector analogy. In vectors, a point
And this is exactly what the directional derivative means: rate of change of Just think of it in this way. When I fly my F-22 through a point in space, I will experience a rate of change of turbulence depending on which direction I choose to travel through that point. So, if I pull up and travel through that point vertically up, the rate of change of turbulence will be different than if I just fly horizontally through it. It all depends on the vector I choose, or the unit vector
While I understand that this lesson is about the gradient vector field, we shall soon see how this links with directional derivative. Right now, just get the meaning of the directional derivative. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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is read as “del phi” and 
is called the del operator. While the geometric meaning of this operator will be elaborated later, just think of it as turning a scalar field to a vector field.
is a function of x and y, in which case
means holding y fixed and differentiating w.r.t to x.
means the gradient of 
evaluated at 
.
is related to the directional derivative of 
. What is this directional derivative? Suppose we are given a point 
in the direction of 
specifying a direction from 
. The directional derivative of 
at 
in the direction of 
is the rate of change of 
with respect to 
as 
varies in the direction of 
from 
. We denote this directional derivative as 
. The reason for the underlining is because the definition needs to have all three terms, 
, 
and 
. In order to under this a bit better, I will compare it with the derivative as we know in 1 variable calculus. 
measures the rate of change of 
was we vary 
. And in the number line, 
can be varied in 2 ways, approaching it from the positive side or from the negative side as illustrated.
in space is specificed by 
and the function we apply is 
. We are to measure the rate of change of 
but this time, in 3-dimensional space, we approach and pass through 
using a vector, a vector which can originate from a variety of ways in the space and NOT limited to a number line like before. We designate this vector, unit vector 
, as illustrated 
with respect to 
as 
varies in the direction of 
from 
. 
.
