Vector Calculus · Engineering Mathematics · GATE ME
Marks 1
The value of the surface integral
where S is the external surface of the sphere x2 + y2 + z2 = R2 is
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)

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 n̂ is the outward unit normal vector on S, is
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 _______.
Marks 2
Consider two vectors
$\rm \vec a = 5 i + 7 j + 2 k $
$\rm \vec b = 3i - j + 6k$
Magnitude of the component of $\vec a$ orthogonal to $\vec b$ in the plane containing the vectors $\vec a$ and $\vec{\bar b}$ is ______ (round off to 2 decimal places).
The value of the integral is _________.
Here, $$\,\,\overrightarrow F x,y = y\widehat i + 2x\widehat j\,\,$$ and $$\,\overrightarrow r $$ is the UNIT tangent vector on the curve $$C$$ at an arc length s from a reference point on the curve. $$\widehat i$$ and $$\widehat j$$ are the basis vectors in the $$X-Y$$ Cartesian reference. In evaluating the line integral, the curve has to be traversed in the counter-clockwise direction.
$$V = \left( {{a_1}x + {a_2}y + {a_3}z} \right)i + \left( {{b_1}x + {b_2}y + {b_3}z} \right)j\,$$ $$ + \left( {{c_1}x + {c_2}y + {c_3}z} \right)k,\,\,$$
where $${{a_1} = 2}$$ and $${{c_3} = - 4.}$$ The value of $${{b_2}}$$ is ________.
Where $$S$$ is the sphere, $$\,\,{x^2} + {y^2} + {z^2} = 1\,\,$$ and $$n$$ is the outward unit normal vector to the sphere. The value of the surface integral is