Vector Calculus · Engineering Mathematics · GATE ME

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Marks 1

GATE ME 2024
The value of the surface integralwhere S is the external surface of the sphere x2 + y2 + z2 = R2 is...
GATE ME 2020 Set 1
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 vec...
GATE ME 2015 Set 3
Let $$\phi $$ be an arbitrary smooth real valued scalar function and $$\overrightarrow V $$ be an arbitrary smooth vector valued function in a three d...
GATE ME 2015 Set 2
Curl of vector $$\,V\left( {x,y,x} \right) = 2{x^2}i + 3{z^2}j + {y^3}k\,\,$$ at $$x=y=z=1$$ is
GATE ME 2014 Set 2
Curl of vector $$\,\,\overrightarrow F = {x^2}{z^2}\widehat i - 2x{y^2}z\widehat j + 2{y^2}{z^3}\widehat k\,\,$$ is
GATE ME 2014 Set 3
Divergence of the vector field $${x^2}z\widehat i + xy\widehat j - y{z^2}\widehat k\,\,$$ at $$(1, -1, 1)$$ is
GATE ME 2012
For the spherical surface $${x^2} + {y^2} + {z^2} = 1,$$ the unit outward normal vector at the point $$\left( {{1 \over {\sqrt 2 }},{1 \over {\sqrt ...
GATE ME 2008
The divergence of the vector field $$\left( {x - y} \right)\widehat i + \left( {y - x} \right)\widehat j + \left( {x + y + z} \right)\widehat k$$ is
GATE ME 2005
Stokes theorem connects
GATE ME 1996
The expression curl $$\left( {grad\,f} \right)$$ where $$f$$ is a scalar function is
GATE ME 1995
If $$\overrightarrow V $$ is a differentiable vector function and $$f$$ is sufficienty differentiable scalar function then curl $$\left( {f\overrighta...

Marks 2

GATE ME 2017 Set 2
The surface integral $$\int {\int\limits_s {F.ndS} } $$ over the surface $$S$$ of the sphere $${x^2} + {y^2} + {z^2} = 9,$$ where $$\,F = \left( {x...
GATE ME 2017 Set 1
For the vector $$\overrightarrow V = 2yz\widehat i + 3xz\widehat j + 4xy\widehat k,$$ the value of $$\,\nabla .\left( {\nabla \times \overrightarrow...
GATE ME 2016 Set 3
The value of the line integral $$\,\,\oint\limits_C {\overrightarrow F .\overrightarrow r ds,\,\,\,} $$ where $$C$$ is a circle of radius $${4 \over {...
GATE ME 2016 Set 2
A scalar potential $$\,\,\varphi \,\,$$ has the following gradient: $$\,\,\nabla \varphi = yz\widehat i + xz\widehat j + xy\widehat k.\,\,$$ Consider...
GATE ME 2015 Set 3
The value of $$\int\limits_C {\left[ {\left( {3x - 8{y^2}} \right)dx + \left( {4y - 6xy} \right)dy} \right],\,\,} $$ (where $$C$$ is the region bounde...
GATE ME 2015 Set 2
The surface integral $$\,\,\int {\int\limits_s {{1 \over \pi }} } \left( {9xi - 3yj} \right).n\,dS\,\,$$ over the sphere given by $${x^2} + {y^2} + {...
GATE ME 2015 Set 1
The velocity field on an incompressible flow is given by $$V = \left( {{a_1}x + {a_2}y + {a_3}z} \right)i + \left( {{b_1}x + {b_2}y + {b_3}z} \right)...
GATE ME 2014 Set 1
The integral $$\,\,\oint\limits_C {\left( {ydx - xdy} \right)\,\,} $$ is evaluated along the circle $${x^2} + {y^2} = {1 \over 4}\,$$ traversed in co...
GATE ME 2013
The following surface integral is to be evaluated over a sphere for the given steady velocity vector field $$F = xi + yj + zk$$ defined with respect t...
GATE ME 2009
The divergence of the vector field $$\,3xz\widehat i + 2xy\widehat j - y{z^2}\widehat k$$ at a point $$(1,1,1)$$ is equal to
GATE ME 2008
The directional derivative of the scalar function $$f(x, y, z)$$$$ = {x^2} + 2{y^2} + z\,\,$$ at the point $$P = \left( {1,1,2} \right)$$ in the direc...
GATE ME 2007
The area of a triangle formed by the tips of vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is
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