GATE AG 2024 • Q48 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Vector differential operators – del, gradient

Question Statement:

The directional derivative of $u(x, y, z) = x^2y + y^2z$ at the point $(1, 2, 3)$ in the direction $\hat{i} + 2\hat{j} + 2\hat{k}$ is ______

Official Answer Key:

11.20 to 11.30

Step-by-Step Solution & Explanation

Step 1 — gradient: ∂u/∂x = 2xy ; ∂u/∂y = x² + 2yz ; ∂u/∂z = y² ∇u = (2xy) î + (x² + 2yz) ĵ + (y²) k̂ Step 2 — evaluate at (1, 2, 3): ∂u/∂x = 2(1)(2) = 4 ∂u/∂y = 1² + 2(2)(3) = 1 + 12 = 13 ∂u/∂z = 2² = 4 ∇u = 4î + 13ĵ + 4k̂ Step 3 — unit vector along the given direction: a⃗ = î + 2ĵ + 3k̂ ; |a⃗| = √(1 + 4 + 9) = √14 = 3.7417 â = (î + 2ĵ + 3k̂)/√14 Step 4 — directional derivative: D = ∇u · â = (4 + 26 + 12)/√14 = 42/3.7417 = 11.22 SECTION 2: FARM MACHINERY

Practice This in Real CBT Exam Interface

Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.

Launch CBT Mock Test