The differentiation provides the slope of a function. In case a function depends upon more than one variable then to find slope concerning a particular variable we do partial differentiation.

EXAMPLE OF A 3 VARIABLE FUNCTION:(here Fx, Fy, Fz may be constant or variable concerning x, y,z respectively) 

$F=F_{x}\widehat{i}+F_{y}\widehat{j} +F_{z}\widehat{k} $

Mark that inverted triangle like structure is called Del. Operator given as 

$\overrightarrow{\nabla}=\dfrac{\partial }{\partial x}\widehat{i}+\dfrac{\partial }{\partial y}\widehat{j}+\dfrac{\partial }{\partial z}\widehat{k} $
In the case of a differential operator, we have mainly three operators (others are there also but they are derived from these operators).

  1. GRADIENT 
  2. DIVERGENCE 
  3. CURL

GRADIENT

  • Mathematical expression 

$\overrightarrow{\nabla}_{Scalar}=\dfrac{\partial F }{\partial x}\widehat{i}+\dfrac{\partial F}{\partial y}\widehat{j}+\dfrac{\partial F}{\partial z}\widehat{k} $

  • Physical meaning(why we need it?) 

The result of gradient on a scale field is a vector (not unit) which has the direction of maximum change in scalar field direction and magnitude is the slope of the field in that particular direction. 

  • How math and reality matches? 

when we do differentiation Function with respect to x we find slope of F with x direction and give it's direction towards x-axis(taking Fy and Fz constant)
$\dfrac{\partial F }{\partial x}=\dfrac{\partial F_{x}}{\partial x}\widehat{i}+\dfrac{\partial F_{y}}{\partial x}\widehat{j}+\dfrac{\partial F_{z}}{\partial x}\widehat{k} $ 

$ \dfrac{\partial F_{y}}{\partial x} $tends to Zero 0 &
$ \dfrac{\partial F_{z}}{\partial x} $tends to Zero 0 as Fy and Fz is constant with respect to x. 
similarly for y & similarly for z also
And adding all these we get a vector and by reverse moving, we can get that direction of the differential vector(dr) which would give rise to the result vector when taken in denominator wrt dF in the numerator, that is the maximum change direction of F.


DIVERGENCE

  • Mathematical expression 

DEL [DOT] FUNCTION
$\dfrac{\partial }{\partial x}\widehat{i}+\dfrac{\partial }{\partial y}\widehat{j}+\dfrac{\partial }{\partial z}\widehat{k} $

  • Physical meaning(why we need it?) 

Divergence measures the outflow of a vector field from a particular point (x, y, z).

  • How math and reality match? (proof) 

Let's take a cuboide of dx*dy*dz dimension 


Divergence in a cube

We will calculate the outflow of F from that box. The flow of F to the cube through face OAFG :$F_y\Delta z\Delta X$

The flow of F to the cube through face OAGF

$\left( F_{y}\right) \Delta x\Delta z$

The flow of F out of the cube through face EBCD 

$\left( F_{y}+\dfrac{\partial F_{y}}{\partial y}\Delta y\right) \Delta x\Delta z$

Now subtracting one from other in y-direction faces as in one face F is to the inside of cube and other is outside. Result is 

$\left( F_{y}\right) \Delta x\Delta z$-$\left( F_{y}+\dfrac{\partial F_{y}}{\partial y}\Delta y\right) \Delta x\Delta z$

=(-$\left( \dfrac{\partial F_{y}}{\partial y}\right) \Delta x\Delta y\Delta z$)

The negative sign shows that the net flow is out of the cube. 

the previous one was for the y-direction similarly for x-direction we will get

$\left( F_{x}+\dfrac{\partial F_{x}}{\partial }\right) \Delta x\Delta y\Delta z$

And for z-direction :

$\left( F_{z}+\dfrac{\partial F_{z}}{\partial z}\right) \Delta x\Delta y\Delta z$

Now adding all these previous found result for all independent direction we will get. 

$\begin{aligned}\left( \dfrac{\partial F_{x}}{\partial x}+\dfrac{\partial Fy}{\partial y}+\dfrac{\partial F_{2}}{\partial z}\right)  \Delta x\Delta y\Delta z\end{aligned}$

you will be that why last delta x,y,z is coming which was not before in the mathematical expression of divergence. It is coming as we have taken a box of that dimension but in that mathematical expression, we are calculating at a particular point and taking that if a box of 1*1*1 dimension would have the same property as that of point.

DEL[•] F

$\dfrac{\partial }{\partial x}\widehat{i}+\dfrac{\partial }{\partial y}\widehat{j}+\dfrac{\partial }{\partial z}\widehat{k} $


CURL

  • Mathematical expression 

$\overrightarrow{\nabla }\times F=\left( \dfrac{\partial \widehat{i}}{\partial x}+\dfrac{\partial }{\partial y}\widehat{j}+\dfrac{\partial }{\partial z}\widehat{k}\right) \times \overrightarrow{F}$

  • Physical meaning(why we need it?) 

 It provides the rotation of a field around a point. Like visualize the 3 pictures properly
Proof of curl Vector


  • How math and reality matches? 

$\left( \dfrac{\partial }{\partial x}\widehat{i}+\dfrac{\partial }{\partial y}\widehat{i}+\dfrac{\partial }{\partial z}\widehat{i}\right) \times\left( F_{x}\widehat{i}+F_{y}\widehat{j}+F_{z}\widehat{k}\right)$

We will apply a trick here I.e. we will bring out the magnitude of |F| from second factor and change the function like below 

$\left( \dfrac{\partial F_{x}}{\partial x}\widehat{i}+\dfrac{\partial F_{y}}{\partial y}\widehat{j}+\dfrac{\partial F_{z}}{\partial z}\widehat{k}\right) \times \dfrac{\overrightarrow{F}}{\left| \overrightarrow{F}\right| }$

Cross multiplication of gradient vector of F with unit vector to F will give a perpendicular vector to both of these (gradient Of F and F)  and the area of cross multiplication will decide curl of F.