Vector Differentiation part 1



Here in this post we will revise our concept of Vector Calculas (differentiation of vectors). This mathematical tool would help us in expressing certain basic ideas with a great convenience while studying electrodynamics.

DIFFERENTIATION OF VECTORS
Consider a vector function f(u) such that
f(u) = fx(u)i + fy(u)j + fz(u)k
where fx, fy(u) and fz(u) are scalar functions of u and are components of vector f(u) along x, y, and z directions.If we want to find the derivative of f(u) with respect to u we will have to proceed in the similar manner we used to do with ordinary derivatives thus

where df(u)/du is also a vecor.
Thus in cartesian coordinated derivative of vector f(u) is given by

SCALAR AND VECTOR FIELDS

When we talk about fields then in this case a particular scalar or vector quantity is defined not just at a point in the space but it is defined continously throughout some region in space or maybe the entire region in the space. Now a scalar field \phi(x,y,z) assocites a scalar with each point in the region of space under consideration and a vector field f(x,y,z) associates a vector with each point.
In electrodynamics we will come across the cases where variation in scalar and vector fields from one point to is continous and is also differentiable in the particular region of space under consideration.

GRADIENT OF A SCALAR FIELD
Consider a scalar field \phi(x,y,z). This function depends on three variables. Now how would we find the derivative of such functions. If we infinitesimal change dx, dy and dz along x, y, and z axis simultaneously then total differential d\phi of function \phi(x,y,z) is given as

above expression comes from our previous knowledge of partial differentiation.
If we closely examine above equation this could be a result of dot product of two vectors thus,

or,

d\phi =(\bigtriangledown \phi)\cdot (d\mathbf{r})

where



is gradient of \phi(x,y,z) and gradient of a scalar function is a vector quantity as it is the multiplication of a vector by a scalar.
Thus we see that gradient of any scalar field has both magnitude and direction. Again consider the function \phi(x,y,z) then from ordinary calculas any change in this function as discussed above is given by

thus,
d\phi =(\bigtriangledown \phi)\cdot (d\mathbf{r}) = |\bigtriangledown \phi| |d\mathbf{r}|cos\theta
From this we see that d\phi(x,y,z) will be maximum when cos\theta=1 which would be the case when dr would be parallel to \bigtriangledown \phi. Thus function \phi changes maximally when one moves in the direction same as that of gradient. So we can say that the direction of \bigtriangledown \phi is along the greatest increase of \phi and the mahnitude of |\bigtriangledown \phi| gives the slope along that direction.

CONCLUSION:

The gradient \bigtriangledown \phi points in the direction of the maximum increase of function \phi(x,y,z) and the magnitude |\bigtriangledown \phi| gives the slope or rate of increase along the maximal direction.

THE OPERATOR \bigtriangledown
While discussing gradient of a scalar function we find that gradient of any function is given by

or,

where the term in parentheses is called “del”

Del is an vector derivative or vector operator and this operator acts on everything to its right in an expression, until the end of the expression or a closing bracket is reached. There are three ways in which ? can act or operate on a scalar or vector function
1. On a scalar function \phi : \bigtriangledown \phi (the gradient);
2. On a vector function f, via the dot product: \bigtriangledown \cdot \mathbf{v} (the divergence);
3. On a vector function f, via the cross product: \bigtriangledown \times \mathbf{v} (the curl).
Out of these three ways of operation of  \bigtriangledown on any function we have already discussed gradient of a scalar function.

In this post we learned about scalar and vector fields, gradient of scalar fields and  \bigtriangledown operator.

In the next post we’ll learn more about vector differential calculus i.e, in particular we’ll discuss divergence and curl of vector fields.
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