Differential Calculus - Mathematical Models - Physics

“Ordinary” Derivatives

Suppose we have a function o f one variable: f i x ) then the derivative, d f kix tells us how rapidly the function fix) varies when we change the argument x by a tiny amount, dx:

In words: If we change x by an amount dx, then f changes by an amount df the derivative is the proportionality factor. For example in figure (a), the function varies slowly with x, and the derivative is correspondingly small. In figure (b), f increases rapidly with x, and the derivative is large, as we move away from x = 0.
Geometrical Interpretation: The derivative d f ! dx is the slope o f the graph off versus x.

Gradient

Suppose that we have a function of three variables-say, V(x, y, z) in a

This tells us how V changes when we alter all three variables by the infinitesimal amounts dx, dy, dz. Notice that we do not require an infinite number o f derivatives-three will suffice: the partial derivatives along each of the three coordinate directions.
Thus
where is the gradient of V .
 is a vector quantity, with three components.

Geometrical Interpretation of the Gradient

Like any vector, the gradient has magnitude and direction. To determine its geometrical meaning, let’s rewrite

where θ is the angle between  and. Now, i f we fix the magnitudeand search around in various directions (that is, varyθ), the maximum change in V evidently occurs when θ = 0(for then cos θ = 1). That is, for a fixed distancedT is greatest when one move in the same direction asThus:
The gradient  points in the direction of maximum increase of the function F.
Moreover:
The magnitudegives the slope (rate o f increase) along this maximal direction.
Gradient in Spherical polar coordinates V (r,θ,∅)

Gradient in cylindrical coordinates V (r,∅,z)


Example 4: Find the gradient of a scalar function of position V where V(x,y, z) = x²y + e². Calculate the magnitude of the gradient at point P(1, 5,-2).

Example 5: Find the unit vector normal to the curve = x² at the point (2, 4, 1).

The equation of curve in the form of surface is given by
x² - y = 0
A constant scalar function V on the surface is given by V (x,y,z) =x² -y
Taking the gradient

The value of the gradient at point (2, 4, 1),
The unit vector, as required

Example 6: Find the unit vector normal to the surface xy³z² = 4 at a point (-1, -1, 2).

Unit vector normal to the surface

Example 7: In electrostatic field problems, the electric field is given b y , where F V s the scalar field potential. If V = r²∅ - 2θ in spherical coordinates, then find

F = r²∅ - 2θ
In spherical coordinate,
Substituting the suitable values,

The Operator

The gradient has the formal appearance of a vector,, “multiplying” a scalar V:
The term in parentheses is called “del”:

We should say that  is a vector operator that acts upon V, not a vector that multiplies V.There are three ways the operatorcan act:
1. on a scalar function V:(the gradient);
2. on a vector function via the dot product: (the divergence),
3. on a vector function  via the cross product: (the curl).

The Divergence

From the definition of  we construct the divergence:

Observe that the divergence of a vector function is itself a scalar (You can't have the divergence of a scalar: that’s meaningless.)

Geometrical Interpretation
is a measure of how much the vector spreads out (diverges) from the point in question. For example, the vector function in figure (a) has a large (positive) divergence (if the arrows pointed in, it would be a large negative divergence), the function in figure (b) has zero divergence, and the function in figure (c) again has a positive divergence. (Please understand that here is a function-there’s a different vector associated with every point in space.)
(a)

(b)

(c)

Divergence in Spherical polar coordinates

Divergence in cylindrical coordinates


Example 8: Suppose the function sketched in above figure are and Calculate their divergences.

Example 9: Given
(i)
(ii)
(iii)

(i) In Cartesian coordinates,
(ii) In cylindrical coordinates 

(iii) In spherical coordinates, 

The Curl

Geometrical Interpretation
 is a measure of how much the vector  ‘‘curls around” the point in question. Figure shown below have a substantial curl, pointing in the z-direction, as the natural right-hand rule would suggest.
(a)

(b)

Curl in Spherical polar coordinates
Curl in cylindrical coordinates

Example 10: Suppose the function sketched in above figure are  and

Calculate their curls.

As expected, these curls point in the +z direction. (Incidentally, they both have zero divergence, as you might guess from the pictures: nothing is “spreading out”.... it just “curls around.”)

Example 11: Given a vector function
(a) Calculate the value of constants c₁,c₂,c₃ if  is irrotational.
(b) Determine the constant c₄ if  is also solenoidal.
(c) Determine the scalar potential function V, whose negative gradient equals.

(a) If A is irrotational then,

(b) Ifis solenoidal,

(c) 

Examination of above expressions of V gives a general value of

Example 12: Find the curl of the vector = (e^-r/r)

= (e^-r/r)⇒A_r = 0, A_θ = (e^-r/r), A_Ф =0

Example 13: Find the nature of the following fields by determining divergence and curl.

(i)
(ii)  (Cylindrical coordinates)

(i)
Divergence exists, so the field is non-solenoidal.
The field has a curl so it is rotational.
(ii)
In cylindrical coordinates, Divergence The field is non-solenoid.

Product Rules

The calculation of ordinary derivatives is facilitated by a number of general rules, such as
the sum rule:

the rule for multiplying by a constant:

the product rule:

and the quotient rule:

Similar relations hold for the vector derivatives. Thus,


and

as you can check for yourself. The product rules are not quite so simple. There are two ways to construct a scalar as the product of two functions:
f g (product of two scalar functions),
(Dot product of two vectors),
and two ways to make a vector:
(Scalar time’s vector),
(Cross product of two vectors),
Accordingly, there are six product rules,
Two for gradients
(i)
(ii)

Two for divergences
(iii)
(iv)
And two for curls
(v)
(vi)
It is also possible to formulate three quotient rules:

Second Derivatives

The gradient, the divergence, and the curl are the only first derivatives we can make with by applyingtwice we can construct five species of second derivatives. The gradient  is a vector, so we can take the divergence and curl of it:
(1) Divergence of gradient:
This object, which we write  for short, is called the Laplacian of V. Notice that the Laplacian o f a scalar V is a scalar.
Laplacian in Spherical polar coordinates

Laplacian in cylindrical coordinates

Occasionally, we shall speak o f the Laplacian o f a vector, By this we mean a vector quantity whose x-component is the Laplacian of A_x, and so on:

This is nothing more than a convenient extension of the meaning of .
(2) Curl of gradient:
The divergence A is a scalar-all we can do is taking its gradient.
The curl of a gradient is always zero:
(3) Gradient of divergence:
The curl  A is a vector, so we can take its divergence and curl.
Notice thatis not the same as the Laplacian of a vector:

(4) Divergence of curl:
 The divergence of a curl, like the curl of a gradient, is always zero:

(5) Curl of curl:
As you can check from the definition of  
So curl-of-curl gives nothing new; the first term is just number (3) and the second is the Laplacian (of a vector).