Questions
1 question per paper
Difficulty
Medium
Importance
Moderate yield for PSU technical papers
Overview
Vector Analysis is a foundational mathematical tool in engineering physics and electromagnetics, focusing on the behavior of scalar and vector fields. In PSU exams, this topic tests your ability to apply differential operators and integral theorems to calculate flux, circulation, and potential fields. Mastering this section is essential as it forms the bedrock for solving complex problems in Electromagnetics and Fluid Mechanics.
Differential Operators: Gradient, Divergence, and Curl
These operators describe the spatial variation of scalar and vector fields. Gradient converts a scalar field to a vector, while Divergence and Curl quantify the expansion and rotation of vector fields respectively.
- Gradient of f: grad(f) = del(f) = (df/dx)i + (df/dy)j + (df/dz)k
- Divergence: div(A) = del dot A = dA_x/dx + dA_y/dy + dA_z/dz
- Curl: curl(A) = del cross A = determinant notation with i, j, k
- Laplacian: del^2(f) = del dot grad(f)
- Solenoidal vector if div(A) = 0
- Irrotational vector if curl(A) = 0
Integral Theorems
The integral theorems provide essential shortcuts for transforming between line, surface, and volume integrals. For PSU exams, identifying the correct theorem to apply is more important than the brute-force integration itself.
- Gauss Divergence Theorem: surface integral of A dot n dS = volume integral of div(A) dV
- Stokes' Theorem: line integral of A dot dr = surface integral of (curl A) dot n dS
- Green's Theorem in Plane: line integral (Pdx + Qdy) = double integral (dQ/dx - dP/dy) dA
- Green's identities connect Laplacian and surface potentials
Vector Identities and Properties
Memorizing specific identities simplifies complex derivations and helps eliminate options in objective-type questions quickly. Focus on identities involving the Laplacian and nested operators.
- div(curl A) = 0
- curl(grad f) = 0
- div(f A) = f(div A) + A dot (grad f)
- curl(f A) = f(curl A) + (grad f) cross A
- del cross (del cross A) = grad(div A) - del^2 A
Formula Sheet
grad(f) = (del f / del x) i + (del f / del y) j + (del f / del z) k
div(A) = del dot A = (del A_x / del x) + (del A_y / del y) + (del A_z / del z)
curl(A) = del cross A = det([i, j, k], [del/del x, del/del y, del/del z], [A_x, A_y, A_z])
Gauss Theorem: integral_S (A dot n) dS = integral_V (del dot A) dV
Stokes Theorem: integral_C (A dot dr) = integral_S (del cross A) dot dS
del dot (del cross A) = 0
del cross (del f) = 0
Exam Tip
Always check if the vector field is solenoidal or irrotational first, as it often reduces the integral to zero or simplifies the derivative terms immediately.
Common Mistakes
- Confusing the direction of the normal vector in surface integrals leading to a sign error.
- Applying Stokes' theorem when the surface is not open or Gauss' theorem when the volume is not closed.
- Forgetting that the divergence of a curl is always zero, leading to unnecessary long-form calculations.
More Revision Notes
Ready to test yourself?
Play topic-wise Vector Analysis questions in Aspirant Arcade — gamified MCQ practice.
Download Free