Vector Analysis

Why it matters

Vector analysis is essential in electrical engineering as it provides the mathematical framework to describe and analyze electromagnetic fields. Understanding vector operations helps in solving complex problems related to electric and magnetic fields, which are fundamental in designing electrical devices and systems.

Key ideas

• Vectors: Quantities having both magnitude and direction, represented in a coordinate system.
• Vector Operations: Include addition, subtraction, dot product, cross product, and scalar multiplication.
• Gradient: Measures the rate and direction of change in a scalar field.
• Divergence: Represents signed local source or sink strength at a given point in a vector field.
• Curl: Describes the rotation of a vector field.
• Coordinate Systems: Cartesian, cylindrical, and spherical systems are used to describe vectors in different contexts.

The component formulas below use Cartesian coordinates and a fixed orthonormal basis. Cylindrical and spherical differential operators require their scale factors.

Formulas

• A + B = (A_x + B_x) i + (A_y + B_y) j + (A_z + B_z) k
  • A, B: Vectors
  • A_x, A_y, A_z, B_x, B_y, B_z: Components of vectors A and B
• A · B = A_x B_x + A_y B_y + A_z B_z
  • A · B: Dot product of vectors A and B
• A × B = (A_y B_z - A_z B_y) i + (A_z B_x - A_x B_z) j + (A_x B_y - A_y B_x) k
  • A × B: Cross product of vectors A and B
• ∇f = (∂f/∂x) i + (∂f/∂y) j + (∂f/∂z) k
  • ∇f: Gradient of scalar field f
• ∇ · A = ∂A_x/∂x + ∂A_y/∂y + ∂A_z/∂z
  • ∇ · A: Divergence of vector field A
• ∇ × A = (∂A_z/∂y - ∂A_y/∂z) i + (∂A_x/∂z - ∂A_z/∂x) j + (∂A_y/∂x - ∂A_x/∂y) k
  • ∇ × A: Curl of vector field A

Worked example

Given: Vector A = 3i + 4j + 5k, Vector B = 2i - j + 3k

1. Calculate the dot product

   • Formula: A · B = A_x B_x + A_y B_y + A_z B_z
   • Calculation: 3*2 + 4*(-1) + 5*3 = 6 - 4 + 15 = 17
   • Dot product: 17

2. Calculate the cross product

   • Formula: A × B = (A_y B_z - A_z B_y) i + (A_z B_x - A_x B_z) j + (A_x B_y - A_y B_x) k
   • Calculation: (4*3 - 5*(-1)) i + (5*2 - 3*3) j + (3*(-1) - 4*2) k = 17i + j - 11k
   • Cross product: 17i + j - 11k

Common mistakes

• Confusing dot product and cross product operations.
• Incorrectly applying vector operations in different coordinate systems.
• Forgetting to consider the direction in vector addition and subtraction.

For GATE EE

• Questions often involve calculating dot and cross products, gradients, divergences, and curls.
• Practice problems in different coordinate systems to strengthen understanding.

Quick check

1. What is the result of the dot product of two perpendicular vectors?
2. How is the gradient of a scalar field represented?
3. What does the divergence of a vector field indicate?

Answers: 1. Zero 2. As a vector 3. Source or sink strength