THE DEL OPERATOR:
Del is not a vector in general.It's actually does not mean much individually untill we provide it some function to act upon.So at first we will understand what's gradient ,then it will be easy for us to understand the Del operator.
WHAT'S GRADIENT:
Let's understand gradient from a example.Suppose,we have a function of three variables say Temperature T(x,y,z) of a room.So if we want to find how the temperature varies from point to point we have to differentiate the function to get the value.Now in this case this function has three variables(x,y,z).So the function not only varies by distance but also it has a direction.It's really difficult to deal with functions with more than one variables .So to generalize the notion of derivative to functions like Temperature we'll use the theorem on partial derivative which states that :
dT=(∂T/∂x)dx+(∂T/∂y)dy+(∂T/∂z)dz ............(1)
this tells us how T changes when we alter all three variables by infinitesimal amounts dx,dy,dz.
we can rearrange equation (1) and we'll get,
dT=[(∂T/∂x) i^+(∂T/∂y) j^+(∂T/∂z) k^].(dx i^+dy j^+dz k^) [ i^, j^, k^ are unit vectors in respective directions]
=(∇T).(dl) [where dl is a vector]
where,
∇T=(∂T/∂x) i^+(∂T/∂y) j^+(∂T/∂z) k^........(2)
is called the Gradient of T.Here,∇T is a vector quantity.
Properties of ∇:
From equation (2), the term in parentheses is called Del:
∇=(∂/∂x) i^ +(∂/∂y) j^+(∂/∂y) k^
We have learnt earlier that ∇ itself is not a vector.But it mimics the behaviour of an ordinary vector in every way.That means we can do almost any sort of application of it so as the vector.
dT=(∂T/∂x)dx+(∂T/∂y)dy+(∂T/∂z)dz ............(1)
this tells us how T changes when we alter all three variables by infinitesimal amounts dx,dy,dz.
we can rearrange equation (1) and we'll get,
dT=[(∂T/∂x) i^+(∂T/∂y) j^+(∂T/∂z) k^].(dx i^+dy j^+dz k^) [ i^, j^, k^ are unit vectors in respective directions]
=(∇T).(dl) [where dl is a vector]
where,
∇T=(∂T/∂x) i^+(∂T/∂y) j^+(∂T/∂z) k^........(2)
is called the Gradient of T.Here,∇T is a vector quantity.
Properties of ∇:
From equation (2), the term in parentheses is called Del:
∇=(∂/∂x) i^ +(∂/∂y) j^+(∂/∂y) k^
We have learnt earlier that ∇ itself is not a vector.But it mimics the behaviour of an ordinary vector in every way.That means we can do almost any sort of application of it so as the vector.
HOW THE OPERATOR ∇ WORKS:
Like an ordinary vector we can use ∇ in three different ways:
1. On a scalar function T: ∇T (the gradient);
2. on a vector function v,by dot product: ∇.v (the divergence);
3. on a vector function v,by cross product: ∇×v (the curl);
You can learn more about operator and it's application in the below e-books: click on the book icon;
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