Vector Analysis

Vector Algebra is a part of algebra that deals with the theory of vectors and vector spaces. Vector analysis is a mathematical tool with which electromagnetic (EM) concepts are most conveniently expressed. It's important to learn its rules and techniques first applying it.

VECTOR ALGEBRA

- Scalars and Vectors.

- Unit Vectors.

- Position and Distance Vectors.


- Vector Multiplication.

- Components Of a Vector.

- Numericals / Solved Exercise.


SOLVED PROBLEMS/NUMERICALS ON VECTOR ALGEBRA:

Q.1. Given vectors A = a_x + 3a_z and B = 5a_x + 2a_y -6a_z, determine

a) |A+B|
B) 5A – B
c) Component of A along a_y.
d) Unit vector parallel to 3A + B.                  SOLUTION/ANSWER


Q.2 Given Points P (1, -3, 5), Q (2, 4, 6) and R (0, 3, 8) find
a) Position vectors of P and R.
b) Distance vector r_QR.
c) Distance between Q and R.                       SOLUTION/ANSWER


Q. 3 Find the unit vector along the line joining point (2, 4, 4) to point (-3, 2, 2) ?                         SOLUTION/ANSWER


Q. 4 Given that A = 3a_x + 5a_y – 7a_z and B = a_x – 2a_y + a_z ; find
a) | 2B + 0.4A |
b) A.B - | B |²
c) A x B                                                           SOLUTION/ANSWER


Q.5 Given Vectors T = 2a_x – 6a_y + 3a_z and S = a_x + 2a_y +az; find
a) the scalar projection of T on S.
b) the vector projection of S on T.
c) the smaller angle between T and S.           SOLUTION/ANSWER


Q.6 Let E = 3a_y + 4a_z and F = 4a_x – 10a_y + 5a_z
a) Find the component of E along F.
b) Determine a Unit vector perpendicular to both E and F.     SOLUTION/ANSWER


Q.7 E and F are vector fields given by E = 2xa_x + a_y +yza_z and F = xya_x – y2a_y +xyza_z.

Determine:
a) | E| at (1, 2, 3)
b) Component of E along F at (1, 2, 3)
c) A vector perpendicular to both E and F at (0,1, -3) whose magnitude is unity?                                                     SOLUTION/ANSWER

Coordinate Systems & Transformation

In general, the physical quantities in ElectroMagnetics are functions of space and time. In order to describe the spatial variations of the quantities, its important to define all points uniquely in space in a suitable manner. This requires using an appropriate coordinate system. Hence its very important to understand the coordinate system first.

Coordinate system is a system of representing points in a space of given dimensions by coordinates, such as the Cartesian coordinate system or the system of celestial longitude and latitude.

Following are the links which will throw more light on the sub-topics related to co-ordinate systems in 3D and transformations:


- Introduction To Coordinate System.

- Cartesian Coordinate System / Rectangular Coordinate System (x, y, z).

- Differential Analysis Of Cartesian Coordinate System.

- Circular Cylindrical Coordinate System (ρ, φ, z).

- Differential Analysis Of Cylindrical Coordinate System.


- Spherical Coordinate System ( r, θ , φ).

- Differential Analysis Of Spherical Coordinate System.

- Numericals / Solved Examples - Page 1.

- Numericals / Solved Examples - Page 2.


SOLVED EXAMPLES ON COORDINATE SYSTEM:


Q.1 Express the following points in Cartesian co-ordinate system.
a) P1 (2, 30^o, 5)
b) P2 (4, 30^o, 60^o)                                   SOLUTION/ANSWER


Q.2. Express the point P (1, -4, -3) in cylindrical and spherical co-ordinates? SOLUTION/ANSWER


Q.3
a) If V = XZ – XY +YZ, express V in Cylindrical co-ordinate system.

b) If U = X² + 2Y² +3z², express U in Spherical co- ordinates System.   SOLUTION/ANSWER


Q.4 Transform the vector E = (y² – x²) a_x + xyz a_y + (x² – z²) a_x to cylindrical and spherical system?      SOLUTION/ANSWER


Q.5 Express the vector A = ρ (z² + 1)a_ρ - ρz cosφa_φ in Cartesian co-ordinate system?       SOLUTION/ANSWER


Q.6 Express the vector E =2r sinθ cosφa_r + r cosθ cosφa_θ – r sinφa_φ in Cartesian co-ordinate system?SOLUTION/ANSWER


Q.7 Calculate the distance between the following pair of points?

a) (2, 1, 5) and (6, -1, 2)
b) (3, π/2, -1) and (5, 3π/2, 5)
c) (10, π/4, 3π/4) and (5, π/6, 7π/4)     SOLUTION/ANSWER


Q.8 Find the distance between A (2, π /6, 0) and B = (1, π /2, 2) ?   SOLUTION/ANSWER


Q.9 Using the differential length dl, for the length of each of the following curves:
a) ρ = 3, π /4 < φ < π /2, z = constant.
b) r = 1, θ = 30^o, 0^o < φ < 60^o
c) r = 4, φ = constant, 30^o < θ < 90^o       SOLUTION/ANSWER


Q.10 Calculate the areas of the following surfaces using the differential surface area ds:
a) ρ = 2, π/3 < φ < π/2, 0 < z < 5.
b) r = 10, π/4 < θ < 2π/3 , 0 < φ < 2π      SOLUTION/ANSWER


Q.11 Use the differential volume dv to determine the volumes of the following regions:
a) 0 < x < 1, 1 < y <2, -3 < z < 3
b) 2 < ρ < 5, π/3 < φ < π, - 1 < z < 4     SOLUTION/ANSWER

VECTOR CALCULUS

Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional space. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow.

Following links will throw more light on the sub-topics:


- Line , Surface and Volume Integral.

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).

- Numericals / Solved Examples - Gradient Of a Scalar.


- Divergence Of a Vector ( ∇ . A ).

- Numericals / Solved Examples - Divergence Of a Vector.

- Curl Of a Vector ( ∇ x A).

- Laplacian Of a Scalar ( ∇² V).


SOLVED NUMERICALS ON VECTOR CALCULUS:

Q.1 Calculate the circulation of A = ρ cosφ a_ρ+ z sinφ a_z around the edge L of the wedge defined by 0 < ρ < 2, 0 < φ < 60^o, z = 0 as shown.








SOLUTION/ANSWER




Q.2 Given that H = x² a_x + y² a_y, evaluate ∫_L H .dl where L is along the curve y =x² from (0, 0) to (1, 1).                    SOLUTION/ANSWER


Q.3 Given that ρ_s = x² + xy, calculate ∫_s ρ_sds over the region y ≤ x², 0< x< 1.               SOLUTION/ANSWER


Q.4 Find the gradient of these scalar fields:
a) U = 4xz₂ + 3yz
b) H = r²cosθ cosφ                SOLUTION/ANSWER


Q.5 If V_{(x, y, z)} = 3x²y –y²z², find ∇ V and |∇ V| at the point (1, 2, -1).   SOLUTION/ANSWER


Q.6 Given φ = xy +yz +xz, find gradient φ at point (1, 2, 3) and the directional derivative of φ.     SOLUTION/ANSWER


Q.7 Find V_{(x, y, z)} if grad V = (y² – 2xyz³)a_x + (3 + 2xy – x²z³)a_y + (4z³ – 3x²yz²)a_z and V_{(0, 0, 0)} = -2.      SOLUTION/ANSWER


Q.8 Find the unit normal vector of the surface x² + y² + z² = 14 at (-1, 3, 2) ?     SOLUTION/ANSWER


Q.9 The temperature in an auditorium is given by T = x² + y² – z. A mosquito located at (1, 1, 2) in the auditorium desires to fly in such a direction that it will get warm as soon as possible. In what direction must it fly?      SOLUTION/ANSWER


Q.10 Determine the divergence of the following vector fields and evaluate them at the specified points:

a) A = yz a_x + 4xy a_y + y a_z at point (1, -2, 3)
b) A = ρzsinφ a_ρ + 3ρz² cosφ a_φ at (5, π / 2, 1)
c) A = 2r cosθ cosφ a_r + r^1/2 a_φ   at (1, π / 6, π / 3)    SOLUTION/ANSWER


Q.11 Determine the flux of D = ρ² cos²φ a_ρ + zsin φ a_φ over the closed surface of the cylinder, 0 < z < 1, ρ =4. Verify the divergence theorem for this case.       SOLUTION/ANSWER


Q.13 Determine the curl of each of the vector field:
a) A = yz a_x + 4xy a_y + y a_z
b) A = ρz sinφ a_ρ + 3ρz²cosφ a_φ             SOLUTION/ANSWER


Q.14 Show that A = (y + z cos xz) a_x + x a_y + xcosxz a_z is conservative, without computing any integrals?                                    SOLUTION/ANSWER


Q.15 Determine the Laplacian of the scalar fields:
a) V = x²y + xyz
b) V = ρz sinφ + z²cos² φ + ρ²                   SOLUTION/ANSWER