Question Bank • QB_EM_ADV_049 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Physical interpretations – line, surface and volume integrals

Question Statement:

The work done by the force field $\mathbf{F} = (2x - y + z)\hat{i} + (x + y - z^2)\hat{j} + (3x - 2y + 4z)\hat{k}$ in moving a particle along the straight line path from $(0, 0, 0)$ to $(1, 1, 1)$ is ________ (round off to two decimal places).

Answer Key:

4.17

Step-by-Step Solution & Explanation

**Method 1: Parametric Line Integration** Parameterize the line segment from $(0,0,0)$ to $(1,1,1)$: $x(t) = t, \quad y(t) = t, \quad z(t) = t, \quad 0 \le t \le 1$ Differentials: $dx = dt, \, dy = dt, \, dz = dt$. Substitute into $\mathbf{F} \cdot d\mathbf{r}$: $\mathbf{F} \cdot d\mathbf{r} = (2t - t + t)\,dt + (t + t - t^2)\,dt + (3t - 2t + 4t)\,dt$ $= 2t\,dt + (2t - t^2)\,dt + 5t\,dt = (9t - t^2)\,dt$ Integrate over $t \in [0, 1]$: $W = \int_0^1 (9t - t^2) \, dt = \left[ \frac{9t^2}{2} - \frac{t^3}{3} \right]_0^1 = \frac{9}{2} - \frac{1}{3} = \frac{27 - 2}{6} = \frac{25}{6} \approx 4.1667 \approx 4.17$
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