Hello everybody. In the last discussion we
have learned vector algebra and now it is
about vector calculus. So, we will start with
differential calculus; obviously, and before
going to vector calculus we would first see
how ordinary derivatives are calculated with
a scalar function.
So, let usconsider differentiation first,
if we have a function f x then its derivative
first order derivative with respect to x that
is f prime x is defined as limit h tends to
0 f x plus h minus f x over h. We are familiar
with this this definition and this in the
shorthand notation is written as df dx.
Geometrically we interpret this derivative
as the slope of the function. Now, if we are
interested in vector differential calculus
then the simplest operation would be gradient
of a scalar field that becomes vector. How
do we define gradient? Let us consider we
have a scalar field T and that is a function
of x, y and z in 3 dimensional Cartesian coordinate
system. So, we can write dT as del T del x
dx plus del T del y dy plus del T del z dz
in the form of partial derivatives.
Now, if wemake a grid we want to take the
gradient of this then the gradient can be
so, dT can be written in the form del T del
x x cap plus del T del y y cap plus del T
del z z cap. This quantity dotted with dx
x cap plus dy y cap plus dz z cap. And we
know that this right hand side is nothing,
but the differential line element; that means,
this quantity becomes the gradient of the
scalar field T that is a function of x, y
and z dotted with the line element dl.
Now, this gradient of T is nothing, but this
quantity here. So, here we define gradient
of T written in this way nabla followed by
T is del T del x x cap plus del T del y y
cap plus del T del z z cap. Geometrically
gradient means the direction in which the
scalar field is changing. So, gradient of
a scalar field is a vector quantity that has
a magnitude and a direction, it is very important
to note.
Let us consider one example of calculating
a gradient, let us try finding the gradient
of the position vector not really vector the
magnitude of the position vector r in 3 dimensional
Cartesian coordinate system. As we know the
definition of gradient is this is given as
del r del x x cap plus del r del y y cap plus
del r del z z cap. And, with the definition
of r that is given here we can clearly see
that the derivative of r with respect to x
will bring us half in front.
And, because there is x square we will have
2 x here and it will the power of r would
be minus half; that means, 1 over x square
plus y square plus z square square root of
this quantity x cap. Similarly, for derivative
with respect to y it will be half 2 y over
x square plus y square plus z square square
root of that y cap and for z it would be half
2 z over the same thing which is the half
and two these things will cancel. And, we
will be left with x x cap plus y y cap plus
z z cap over x squared plus y squared plus
z squared square root of this quantity.
So, the numerator is nothing, but the r vector
over the denominator the magnitude of r; that
means, the gradient of the scalar of the scalar
magnitude of the position vector gives us
the direction of r, that is unit vector along
the position vector r. Let us consider another
example .
Let us consider a scalar field, if that is
a function of all three position coordinates
x, y and z is given as x squared plus y cubed
plus z power 4. Now, if we calculate the gradient
of this scalar field we will have we can write
it as x cap del del x plus y cap del del y
plus z cap del del z. This is the gradient
operator and this operator is operated on
the scalar field that is x square plus y cubed
plus z power 4 .
So, we will get x cap 2 x plus y cap 3 y squared
plus z cap 4 z cubed which is rearranging
it properly twice x x cap plus 3 y squared
y cap plus 4 z cubed z cap .
So, we can see that we have developed a del
operator out of this. The way in the previous
example we have written the gradient of operator
was this operator was equal to x cap del del
x plus y cap del del y plus z cap del del
z. Although this is a differential operator,
it looks pretty much like a vector and just
because it looks like a vector we can define
its product with a scalar, its product with
another vectors that will look like a dot
product or a cross product. But, we have to
remember that it is not really a dot product
and cross product of vectors rather it is
a differential operator, we have to be careful
about that .
