Questions
~2 questions per paper
Difficulty
Medium
Importance
Medium yield for ONGC, NTPC, and HPCL
Overview
Calculus for PSU exams is a focused set of advanced mathematical tools that test your ability to perform spatial analysis and optimization. Mastery of these concepts is essential for solving engineering problems related to field theory, fluid dynamics, and system efficiency in industrial settings.
Mean Value Theorems and Optimization
These theorems bridge the gap between local derivatives and global behavior of functions. In PSU exams, questions focus on locating critical points and verifying if a function meets the criteria for Rolle's or Lagrange's theorem on a given interval.
- Rolle's Theorem: f(a) = f(b) implies f'(c) = 0
- Lagrange's MVT: f'(c) = (f(b)-f(a))/(b-a)
- Maxima/Minima: Set f'(x) = 0 and check f''(x) sign
- Point of Inflection: Where f''(x) changes sign
Vector Calculus: Operators
Vector operators define how scalar and vector fields vary in space. Understanding the physical interpretation of Gradient, Divergence, and Curl is crucial for electromagnetics and fluid flow problems.
- Gradient of scalar field: grad(f) = del(f)
- Divergence of vector field: div(A) = del dot A
- Curl of vector field: curl(A) = del cross A
- Solenoidal field: divergence is zero
- Irrotational field: curl is zero
Integral Theorems
These theorems provide powerful methods to convert between different types of integrals across spatial dimensions. They are highly examinable, often appearing as direct calculation problems.
- Gauss Divergence Theorem: Volume to Surface conversion
- Stokes' Theorem: Surface to Line integral conversion
- Green's Theorem: Relates line integral to double integral in 2D
- Application: Calculate flux through closed surfaces
Formula Sheet
Directional Derivative: del(f) dot n_cap
Laplacian: del^2(f) = div(grad(f))
Gauss Theorem: triple_int(div A)dV = double_int(A dot n_cap)dS
Stokes Theorem: double_int(curl A dot n_cap)dS = line_int(A dot dr)
Green's Theorem: line_int(Mdx + Ndy) = double_int(dN/dx - dM/dy)dA
Exam Tip
Memorize the directional derivative formula as grad(f) dot unit vector, as this is the most frequent shortcut-based question in PSU papers.
Common Mistakes
- Confusing the order of cross products when computing Curl, leading to sign errors.
- Forgetting to check the 'closed surface' condition before applying Gauss Divergence Theorem.
- Misidentifying the limits of integration in triple integrals for volume calculations.
More Revision Notes
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