1
GATE ME 2017 Set 2
Numerical
+2
-0
The surface integral $$\int {\int\limits_s {F.ndS} } $$ over the surface $$S$$ of the sphere $${x^2} + {y^2} + {z^2} = 9,$$ where $$\,F = \left( {x + y} \right){\rm I} + \left( {x + z} \right)j + \left( {y + z} \right)k\,\,$$ and $$n$$ is the unit outward surface normal, yields ___________.
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2
GATE ME 2017 Set 1
Numerical
+2
-0
For the vector $$\overrightarrow V = 2yz\widehat i + 3xz\widehat j + 4xy\widehat k,$$ the value of $$\,\nabla .\left( {\nabla \times \overrightarrow \nabla } \right)\,\,$$ is ______________.
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3
GATE ME 2016 Set 2
Numerical
+2
-0
A scalar potential $$\,\,\varphi \,\,$$ has the following gradient: $$\,\,\nabla \varphi = yz\widehat i + xz\widehat j + xy\widehat k.\,\,$$ Consider the integral $$\,\,\int_C {\nabla \varphi .d\overrightarrow r \,\,} $$ on the curve $$\overrightarrow r = x\widehat i + y\widehat j + z\widehat k.\,\,$$ The curve $$C$$ is parameterized as follows: $$\,\,\left\{ {\matrix{ {x = t} \cr {y = {t^2}} \cr {z = 3{t^2}} \cr } \,\,\,\,\,\,\,} \right.$$ and $$1 \le t \le 3.\,\,\,\,\,$$
The value of the integral is _________.
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4
GATE ME 2016 Set 3
Numerical
+2
-0
The value of the line integral $$\,\,\oint\limits_C {\overrightarrow F .\overrightarrow r ds,\,\,\,} $$ where $$C$$ is a circle of radius $${4 \over {\sqrt \pi }}\,\,$$ units 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.

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