Differential Calculus Notes | Study Mathematical Models - Physics

Physics: Differential Calculus Notes | Study Mathematical Models - Physics

The document Differential Calculus Notes | Study Mathematical Models - Physics is a part of the Physics Course Mathematical Models.
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“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 Notes | Study Mathematical Models - 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 Notes | Study Mathematical Models - Physics

Gradient

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

where θ is the angle between Differential Calculus Notes | Study Mathematical Models - Physics andDifferential Calculus Notes | Study Mathematical Models - Physics. Now, i f we fix the magnitudeDifferential Calculus Notes | Study Mathematical Models - 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 Notes | Study Mathematical Models - PhysicsdT is greatest when one move in the same direction asDifferential Calculus Notes | Study Mathematical Models - PhysicsThus:
The gradient Differential Calculus Notes | Study Mathematical Models - Physics points in the direction of maximum increase of the function F.
Moreover:
The magnitudeDifferential Calculus Notes | Study Mathematical Models - Physicsgives the slope (rate o f increase) along this maximal direction.
Gradient in Spherical polar coordinates V (r,θ,∅)
Differential Calculus Notes | Study Mathematical Models - Physics
Gradient in cylindrical coordinates V (r,∅,z)
Differential Calculus Notes | Study Mathematical Models - 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 Notes | Study Mathematical Models - Physics
Differential Calculus Notes | Study Mathematical Models - 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 Notes | Study Mathematical Models - Physics
The value of the gradient at point (2, 4, 1), Differential Calculus Notes | Study Mathematical Models - Physics
The unit vector, as required
Differential Calculus Notes | Study Mathematical Models - Physics


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

Differential Calculus Notes | Study Mathematical Models - Physics
Differential Calculus Notes | Study Mathematical Models - Physics
Unit vector normal to the surface
Differential Calculus Notes | Study Mathematical Models - Physics


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

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


The Operator Differential Calculus Notes | Study Mathematical Models - Physics

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

The Divergence

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

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

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

Differential Calculus Notes | Study Mathematical Models - Physics
Differential Calculus Notes | Study Mathematical Models - Physics
Differential Calculus Notes | Study Mathematical Models - Physics


Example 9: Given
(i)Differential Calculus Notes | Study Mathematical Models - Physics
(ii)Differential Calculus Notes | Study Mathematical Models - Physics
(iii)Differential Calculus Notes | Study Mathematical Models - Physics

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


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

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


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

Calculate their curls.

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

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


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

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


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

(i) Differential Calculus Notes | Study Mathematical Models - Physics
(ii) Differential Calculus Notes | Study Mathematical Models - Physics (Cylindrical coordinates)

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


Product Rules

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

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

Second Derivatives

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

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