Engineering Mathematics-I Unit 5 – Vector Calculus: important questions for AKTU
Unit 5 (Vector Calculus) questions that AKTU repeats most often. This unit carries about 29 marks per paper. Start with the repeated questions, then the most asked topics.
Most repeated Unit 5 questions
- Show that the vector $f(r)\vec{r}$ is irrotational. Where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. (7 marks, 2025, Gradient, Curl, Divergence and Their Physical Interpretations) – also asked in 2019, 2020, 2021, 2022, 2024, 2026
- Apply Gauss’s divergence theorem to evaluate $\iint_{S} \vec{F} \cdot \hat{n} ds$, where $\vec{F} = (2x^2 - 3z)i - 2xyj - 4xk$ and S is the Surface bounded by the planes $x = 0, y = 0, z = 0$ and $2x + 2y + z = 4$. (7 marks, 2026, Gauss' Divergence Theorem (Without Proof) and Applications) – also asked in 2019, 2020, 2021, 2023, 2024
- Apply Gauss divergence theorem to evaluate $\iint_S \vec{F} \cdot \hat{n} ds$, where $\vec{F} = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k}$ and $S$ is the surface of the region bounded by the cylinder $x^2+y^2=4, z=0, z=3$. (7 marks, 2023, Gauss' Divergence Theorem (Without Proof) and Applications) – also asked in 2019, 2020, 2021, 2024, 2026
Most important Unit 5 topic
- Gradient, Curl, Divergence and Their Physical Interpretations (Unit 5: Vector Calculus) – asked 15 times in 2019, 2020, 2021, 2022, 2023, 2024, 2025, 2026
Most asked Unit 5 topics
- Gradient, Curl, Divergence and Their Physical Interpretations – asked 15 times
- Directional Derivatives – asked 6 times
- Gauss' Divergence Theorem (Without Proof) and Applications – asked 6 times
More Unit 5 previous year questions
- Show that gradient field describing a motion is irrotational. (2 marks, 2026, Gradient, Curl, Divergence and Their Physical Interpretations)
- Find the directional derivative of the divergence of $f(x, y, z) = xyi + xy^2j + z^2k$ at the point (2, 1, 2) in the direction of the outer normal to the sphere $x^2 + y^2 + z^2 = 9$. (7 marks, 2026, Directional Derivatives)
- Apply Green’s theorem to evaluate $\int_{C} [(y - \sin x) dx + \cos x dy]$, where C is the triangle formed by $y = 0, x = \frac{\pi}{2}, y = \frac{2}{\pi}x$. (7 marks, 2026, Green's Theorem (Without Proof) and Applications)
- Prove that $curl \vec{r} = 0$. (2 marks, 2025, Gradient, Curl, Divergence and Their Physical Interpretations)
- Show that $div (grad r^n) = n(n + 1)r^{n-2}$. Where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. (7 marks, 2025, Laplacian and Radial Scalar Fields)
Unit 5 syllabus topics
- Gradient, Curl, Divergence and Their Physical Interpretations
- Directional Derivatives
- Line Integrals
- Surface Integrals
- Volume Integrals
- Gauss' Divergence Theorem (Without Proof) and Applications
- Green's Theorem (Without Proof) and Applications
- Stokes' Theorem (Without Proof) and Applications
- Laplacian and Radial Scalar Fields