Vector Differentiation 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) = f_x(u)i + f_y(u)j + f_z(u)k
where f_x, f_y(u) and f_z(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 φ(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 φ(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 Dφ of function φ(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,
dφ=(∇φ)•(dr)
where




is gradient of φ(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 φ(x,y,z) then from ordinary calculas any change in this function as discussed above is given by




thus
dφ=(∇φ)•(dr) = |∇φ| |dr|cosθ
FRom this we see that dφ(x,y,z) will be maximum when cosθ=1 which would be the case when dr would be parallel to ∇φ. Thus function φ changes maximally when one moves in the direction same as that of gradient. So we can say that the direction of ∇φ is along the greatest increase of φ and the mahnitude of |∇φ| gives the slope along that direction.
CONCLUSION: The gradient ∇φ points in the direction of the maximum increase of function φ(x,y,z) and the magnitude |∇φ| gives the slope or rate of increase along the maximal direction.

THE OPERATOR ∇
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 φ : ∇φ (the gradient);
2. On a vector function f, via the dot product: ∇ • v (the divergence);
3. On a vector function f, via the cross product: ∇ x v (the curl).
Out of these three ways of operation of ∇ 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 ∇ operator. In the next post we'll lern more about vector differential calculas i.e, in particular we'll discuss divergance and curl of vector fields.
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Vector Algebra 2

In this post we'll lern Vector algebra in component form.
Component of any vector is the projection of that vector along the three coordinate axis X, Y, Z.

VECTOR ADDITION
In component form addition of two vectors is
C = (A_x+ B_x)i + (A_y+ B_y)j + (A_y+ B_y)k
where,
A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z)
Thus in component form resultant vector C becomes,
C_x = A_x+ B_x
C_y = A_y+ B_y
C_z = A_z+ B_z

SUBTRACTION OF TWO VECTORS
In component form subtraction of two vectors is
D = (A_x- B_x)i + (A_y- B_y)j + (A_y- B_y)k
where,
A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z)
Thus in component form resultant vector D becomes,
D_x = A_x - B_x
D_y = A_y- B_y
D_z = A_z- B_z

NOTE:- Two vectors add or subtract like components.

DOT PRODUCT OF TWO VECTORS
A.B = (A_xi + A_yj + A_zk) . (B_xi + B_yj + B_zk)
= A_xB_x + A_yB_y + A_zB_z.
Thus for calculating the dot product of two vectors, first multiply like components, and then add.

CROSS PRODUCT OF TWO VECTORS

A x B = (A_xi + A_yj + A_zk) x (B_xi + B_yj + B_zk)
= (A_yB_z - A_zB_y)i + (A_zB_x - A_xB_z)j + ( A_xB_y - A_yB_x)k.

Cross product of two vectors is itself a vector.
To calculate the cross product, form the determinantwhose first row is x, y, z, whose second row is A (in component form), and whose third row is B.

VECTOR TRIPPLE PRODUCT

Vector product of two vectors can be made to undergo dot or cross product with any third vector.

(a) Scalar tripple product:-
For three vectors A, B, and C, their scalar triple product is defined as
A . (B x C) = B . (C x A) = C . (A x B)
obtained in cyclic permutation. If A = (A_x, A_y, A_z) , B = (B_x, B_y, B_z) , and C = (C_x, C_y, C_z) then A . (B x C) is the volume of a parallelepiped having A, B, and C as edges and can easily obtained by finding the determinant of the 3 x 3 matrix formed by A, B, and C.

(b) Vector Triple Product:-
For vectors A, B, and C, we define the vector tiple product as
A x (B x C) = B(A . C) - C(A - B)
Note that
(A . B)C ≠ A(B . C)
but
(A . B)C = C(A . B).

Vector Algebra 1

 Here in this post we will go through a quick recap of vector algebra keeping in mind that reader already had detail knowledge and problem solving skills related to the topic being discussed. Here we are briefing Vector Algebra because concepts of electrostatics , electromagnetism and many more physical phenomenon can best be conveniently expressed using this tool.

A vector is a quantity that requires both a magnitude (= 0) and a direction in space it can be represented by an arrow in space for example electrostatic force, electrostatic field etc. In symbolic form we will represent vectors by bold letters. In component form vector A is written as
A = A_xi+ A_yj+A_zk


ADDITION OF VECTORS
Two vectors A and B can be added together to give another resultant vector C.
C = A + B

SUBTRACTION OF VECTORS
Two vectors A and B can be subtracted to give another resultant vector D.
D = A - B = A + (-B)

SCALAR MULTIPLICATION OF VECTOR
When we multiply any vector A with any scalar quantity 'n' then it's direction remains unchanged and magnitude gets multiplied by 'n'. Thus,
n(A) = nA
Scalar multiplication of vectors is distributive i.e.,
n(A + B) = nA +nB

DOT PRODUCT OF VECTORS
Dot product of two vectors A and B is defined as the product of the magnitudes of vectors A and B and the cosine of the angle between them when both te vectors are placed tail to tail. Dot product is represented as A.B thus,
A.B = |A| |B| cosθ
where θ is the angle between two vectors.
Result of dot product of two vectors is a scalar quantity.
Dot product is commutative : A.B = B.A
Dot product is distributive : A . (B+C) = A.B + A.C also A.A = |A|²

CROSS PRODUCT OF TWO VECTORS
Cross product or vector product of two vectors A and B is defined as
A x B = |A| |B| sinθ nˆ
where nˆ is the unit vector pointing in the direction perpandicular to the plane of both A and B. Result of vector product is also a vector quantity.
Cross product is distributive i.e., A x (B + C) = (A x B) + (A x C) but not commutative and the cross product of two parallel vectors is zero.

In our next post we'll study vector algebra in component form and also lern about vector triple products.