1
GATE ME 2023
Numerical
+1
-0

A vector field

𝐁(𝑥, 𝑦, 𝑧) = 𝑥 𝑖̂ + 𝑦 ĵ − 2𝑧 k̂

is defined over a conical region having height ℎ = 2, base radius 𝑟 = 3 and axis along z, as shown in the figure. The base of the cone lies in the x-y plane and is centered at the origin.

If 𝒏 denotes the unit outward normal to the curved surface 𝑆 of the cone, the value of the integral

$\rm \int_SB.n\ dS$

equals _________ . (Answer in integer)

GATE ME 2023 Engineering Mathematics - Vector Calculus Question 1 English
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2
GATE ME 2022 Set 2
MCQ (Single Correct Answer)
+1
-0.33
Consider a cube of unit edge length and sides parallel to co-ordinate axes, with its centroid at the point (1, 2, 3). The surface integral $\int_A \vec{F}.d\vec{A}$ of a vector field $\vec{F}=3x\hat{i}+5y\hat{j}+6z\hat{k}$ over the entire surface A of the cube is ______.
A
14
B
27
C
28
D
31
3
GATE ME 2022 Set 1
MCQ (Single Correct Answer)
+1
-0.33

Given a function $\rm ϕ = \frac{1}{2} (x^2 + y^2 + z^2) $ in three-dimensional Cartesian space, the value of the surface integral

S n̂ . ∇ϕ dS

where S is the surface of a sphere of unit radius and is the outward unit normal vector on S, is

A
B
C
4π/3
D
0
4
GATE ME 2020 Set 1
Numerical
+1
-0

For three vectors $$\vec A = 2\hat j - 3\hat k,\vec B = - 2\hat i + \hat k\ and\;\vec C = 3\hat i - \hat j,$$ where î, ĵ and k̂ are unit vectors along the axes of a right-handed rectangular/Cartesian coordinate system, the value of $$\left( {\vec {A.} \left( {\vec B \times \vec C} \right) + 6} \right)$$ is _______.

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