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Differential Calculus | Mathematical Methods - Physics PDF Download

“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:
Differential Calculus | Mathematical Methods - Physics
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.
Differential Calculus | Mathematical Methods - Physics

Gradient

Suppose that we have a function of three variables-say, V(x, y, z) in a
Differential Calculus | Mathematical Methods - Physics
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 Differential Calculus | Mathematical Methods - Physics
where Differential Calculus | Mathematical Methods - Physicsis the gradient of V .
Differential Calculus | Mathematical Methods - Physics 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
Differential Calculus | Mathematical Methods - Physics

where θ is the angle between Differential Calculus | Mathematical Methods - Physics andDifferential Calculus | Mathematical Methods - Physics. Now, i f we fix the magnitudeDifferential Calculus | Mathematical Methods - Physicsand 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 distanceDifferential Calculus | Mathematical Methods - PhysicsdT is greatest when one move in the same direction asDifferential Calculus | Mathematical Methods - PhysicsThus:
The gradient Differential Calculus | Mathematical Methods - Physics points in the direction of maximum increase of the function F.
Moreover:
The magnitudeDifferential Calculus | Mathematical Methods - Physicsgives the slope (rate o f increase) along this maximal direction.
Gradient in Spherical polar coordinates V (r,θ,∅)
Differential Calculus | Mathematical Methods - Physics
Gradient in cylindrical coordinates V (r,∅,z)
Differential Calculus | Mathematical Methods - Physics

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).

Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics


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
Differential Calculus | Mathematical Methods - Physics
The value of the gradient at point (2, 4, 1), Differential Calculus | Mathematical Methods - Physics
The unit vector, as required
Differential Calculus | Mathematical Methods - Physics


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

Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics
Unit vector normal to the surface
Differential Calculus | Mathematical Methods - Physics


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

F = r2∅ - 2θ
In spherical coordinate, Differential Calculus | Mathematical Methods - Physics
Substituting the suitable values, Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics


The Operator Differential Calculus | Mathematical Methods - Physics

The gradient has the formal appearance of a vector,Differential Calculus | Mathematical Methods - Physics, “multiplying” a scalar V:Differential Calculus | Mathematical Methods - Physics
The term in parentheses is called “del”:
Differential Calculus | Mathematical Methods - Physics
We should say that Differential Calculus | Mathematical Methods - Physics is a vector operator that acts upon V, not a vector that multiplies V.There are three ways the operatorDifferential Calculus | Mathematical Methods - Physicscan act:
1. on a scalar function V:Differential Calculus | Mathematical Methods - Physics(the gradient);
2. on a vector function Differential Calculus | Mathematical Methods - Physicsvia the dot product: Differential Calculus | Mathematical Methods - Physics(the divergence),
3. on a vector function Differential Calculus | Mathematical Methods - Physics via the cross product: Differential Calculus | Mathematical Methods - Physics(the curl).

The Divergence

From the definition of Differential Calculus | Mathematical Methods - Physics we construct the divergence:
Differential Calculus | Mathematical Methods - Physics
Observe that the divergence of a vector function Differential Calculus | Mathematical Methods - Physicsis itself a scalar Differential Calculus | Mathematical Methods - Physics(You can't have the divergence of a scalar: that’s meaningless.)

Geometrical Interpretation
Differential Calculus | Mathematical Methods - Physicsis a measure of how much the vector Differential Calculus | Mathematical Methods - Physicsspreads 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 Differential Calculus | Mathematical Methods - Physicshere is a function-there’s a different vector associated with every point in space.)
(a)
Differential Calculus | Mathematical Methods - Physics
(b)
Differential Calculus | Mathematical Methods - Physics
(c)
Differential Calculus | Mathematical Methods - Physics
Divergence in Spherical polar coordinates
Differential Calculus | Mathematical Methods - Physics
Divergence in cylindrical coordinates
Differential Calculus | Mathematical Methods - Physics

Example 8: Suppose the function sketched in above figure are Differential Calculus | Mathematical Methods - Physicsand Differential Calculus | Mathematical Methods - PhysicsCalculate their divergences.

Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics


Example 9: Given
(i)Differential Calculus | Mathematical Methods - Physics
(ii)Differential Calculus | Mathematical Methods - Physics
(iii)Differential Calculus | Mathematical Methods - Physics

(i) In Cartesian coordinates,Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics(ii) In cylindrical coordinates Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics
(iii) In spherical coordinates, Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics


The Curl


From the definition of Differential Calculus | Mathematical Methods - Physics we construct the curl
Differential Calculus | Mathematical Methods - PhysicsDifferential Calculus | Mathematical Methods - Physics
Notice that the curl of a vector function Differential Calculus | Mathematical Methods - Physics is, like any cross product, a vector. (You cannot have the curl of a scalar; that’s meaningless.)

Geometrical Interpretation
Differential Calculus | Mathematical Methods - Physics is a measure of how much the vector Differential Calculus | Mathematical Methods - Physics ‘‘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)
Differential Calculus | Mathematical Methods - Physics
(b)
Differential Calculus | Mathematical Methods - Physics
Curl in Spherical polar coordinatesDifferential Calculus | Mathematical Methods - Physics
Curl in cylindrical coordinates Differential Calculus | Mathematical Methods - Physics


Example 10: Suppose the function sketched in above figure are Differential Calculus | Mathematical Methods - Physics and Differential Calculus | Mathematical Methods - Physics

Calculate their curls.

Differential Calculus | Mathematical Methods - PhysicsAs 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 Differential Calculus | Mathematical Methods - Physics
(a) Calculate the value of constants c1,c2,c3 if Differential Calculus | Mathematical Methods - Physics is irrotational.
(b) Determine the constant c4 if Differential Calculus | Mathematical Methods - Physics is also solenoidal.
(c) Determine the scalar potential function V, whose negative gradient equalsDifferential Calculus | Mathematical Methods - Physics.

(a) If A is irrotational then,Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics
(b) IfDifferential Calculus | Mathematical Methods - Physicsis solenoidal,
Differential Calculus | Mathematical Methods - Physics
(c) Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics
Examination of above expressions of V gives a general value of
Differential Calculus | Mathematical Methods - Physics


Example 12: Find the curl of the vector Differential Calculus | Mathematical Methods - Physics= (e-r/r)Differential Calculus | Mathematical Methods - Physics

Differential Calculus | Mathematical Methods - Physics= (e-r/r)Differential Calculus | Mathematical Methods - Physics⇒A= 0, Aθ = (e-r/r), AФ =0
Differential Calculus | Mathematical Methods - Physics


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

(i) Differential Calculus | Mathematical Methods - Physics
(ii) Differential Calculus | Mathematical Methods - Physics (Cylindrical coordinates)

(i) Differential Calculus | Mathematical Methods - Physics
Divergence exists, so the field is non-solenoidal.
Differential Calculus | Mathematical Methods - PhysicsThe field has a curl so it is rotational.
(ii) Differential Calculus | Mathematical Methods - Physics
In cylindrical coordinates, Divergence Differential Calculus | Mathematical Methods - PhysicsThe field is non-solenoid.
Differential Calculus | Mathematical Methods - Physics


Product Rules

The calculation of ordinary derivatives is facilitated by a number of general rules, such as
the sum rule:
Differential Calculus | Mathematical Methods - Physics
the rule for multiplying by a constant:
Differential Calculus | Mathematical Methods - Physics
the product rule:
Differential Calculus | Mathematical Methods - Physics
and the quotient rule:
Differential Calculus | Mathematical Methods - Physics
Similar relations hold for the vector derivatives. Thus,
Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - Physics
and
Differential Calculus | Mathematical Methods - Physics
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),
Differential Calculus | Mathematical Methods - Physics(Dot product of two vectors),
and two ways to make a vector:
Differential Calculus | Mathematical Methods - Physics(Scalar time’s vector),
Differential Calculus | Mathematical Methods - Physics(Cross product of two vectors),
Accordingly, there are six product rules,
Two for gradients
(i) Differential Calculus | Mathematical Methods - Physics
(ii) Differential Calculus | Mathematical Methods - Physics

Two for divergences
(iii)Differential Calculus | Mathematical Methods - Physics
(iv)Differential Calculus | Mathematical Methods - Physics
And two for curls
(v)Differential Calculus | Mathematical Methods - Physics
(vi)Differential Calculus | Mathematical Methods - Physics
It is also possible to formulate three quotient rules:
Differential Calculus | Mathematical Methods - Physics

Second Derivatives

The gradient, the divergence, and the curl are the only first derivatives we can make with Differential Calculus | Mathematical Methods - Physicsby applyingDifferential Calculus | Mathematical Methods - Physicstwice we can construct five species of second derivatives. The gradient Differential Calculus | Mathematical Methods - Physics is a vector, so we can take the divergence and curl of it:
(1) Divergence of gradient: Differential Calculus | Mathematical Methods - Physics
Differential Calculus | Mathematical Methods - PhysicsThis object, which we write Differential Calculus | Mathematical Methods - Physics 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
Differential Calculus | Mathematical Methods - Physics
Laplacian in cylindrical coordinates
Differential Calculus | Mathematical Methods - Physics
Occasionally, we shall speak o f the Laplacian o f a vector, Differential Calculus | Mathematical Methods - PhysicsBy this we mean a vector quantity whose x-component is the Laplacian of Ax, and so on:
Differential Calculus | Mathematical Methods - Physics
This is nothing more than a convenient extension of the meaning of Differential Calculus | Mathematical Methods - Physics.
(2) Curl of gradient: Differential Calculus | Mathematical Methods - Physics
The divergence Differential Calculus | Mathematical Methods - PhysicsA is a scalar-all we can do is taking its gradient.
The curl of a gradient is always zero:Differential Calculus | Mathematical Methods - Physics
(3) Gradient of divergence:Differential Calculus | Mathematical Methods - Physics
The curl Differential Calculus | Mathematical Methods - Physics A is a vector, so we can take its divergence and curl.
Notice thatDifferential Calculus | Mathematical Methods - Physicsis not the same as the Laplacian of a vector:
Differential Calculus | Mathematical Methods - Physics
(4) Divergence of curl:Differential Calculus | Mathematical Methods - Physics
 The divergence of a curl, like the curl of a gradient, is always zero:
Differential Calculus | Mathematical Methods - Physics
(5) Curl of curl: Differential Calculus | Mathematical Methods - Physics
As you can check from the definition of Differential Calculus | Mathematical Methods - Physics 
So curl-of-curl gives nothing new; the first term is just number (3) and the second is the Laplacian (of a vector).

The document Differential Calculus | Mathematical Methods - Physics is a part of the Physics Course Mathematical Methods.
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FAQs on Differential Calculus - Mathematical Methods - Physics

1. What is differential calculus?
Ans. Differential calculus is a branch of mathematics that deals with the study of rates of change and slopes of curves. It focuses on understanding how quantities change with respect to independent variables. It involves concepts such as derivatives, differentiation, and finding the rate of change of a function at a specific point.
2. What is the IIT JAM exam?
Ans. The IIT JAM (Joint Admission Test for M.Sc.) is a national-level entrance exam conducted by the Indian Institutes of Technology (IITs) for admission to various postgraduate programs in science. It is a highly competitive exam that tests the candidates' knowledge and understanding of subjects such as mathematics, physics, chemistry, biotechnology, etc.
3. How can I prepare for differential calculus in the IIT JAM exam?
Ans. To prepare for differential calculus in the IIT JAM exam, it is essential to have a strong understanding of the fundamental concepts and techniques. Here are a few tips to prepare effectively: - Start by thoroughly studying the basic concepts of differential calculus, such as limits, derivatives, and their properties. - Practice solving a variety of problems from textbooks, previous year question papers, and online resources to enhance your problem-solving skills. - Understand the application of differential calculus in real-life scenarios, such as optimization problems and rates of change. - Take mock tests and solve sample papers to get familiar with the exam pattern and time management.
4. What are some important topics in differential calculus for the IIT JAM exam?
Ans. Some important topics in differential calculus for the IIT JAM exam include: - Limits and continuity - Differentiation of algebraic, trigonometric, exponential, and logarithmic functions - Chain rule and implicit differentiation - Higher-order derivatives and Leibnitz theorem - Applications of derivatives, such as maxima and minima, tangents, and normals
5. Are there any recommended books or study materials for differential calculus in the IIT JAM exam?
Ans. Yes, there are several recommended books and study materials for differential calculus in the IIT JAM exam. Some popular choices include: - "Calculus" by Michael Spivak - "Differential Calculus" by Shanti Narayan - "Differential Calculus" by Amit M. Agarwal - "Differential Calculus for IIT JAM" by Sanjay Mishra - Practice sets and solved papers by Arihant Publications and Wiley Publications.
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