GATE ME
Engineering Mathematics
Vector Calculus
Previous Years Questions

## Marks 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...
Let $$\phi$$ be an arbitrary smooth real valued scalar function and $$\overrightarrow V$$ be an arbitrary smooth vector valued function in a three d...
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
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
Divergence of the vector field $${x^2}z\widehat i + xy\widehat j - y{z^2}\widehat k\,\,$$ at $$(1, -1, 1)$$ is
For the spherical surface $${x^2} + {y^2} + {z^2} = 1,$$ the unit outward normal vector at the point $$\left( {{1 \over {\sqrt 2 }},{1 \over {\sqrt ... The divergence of the vector field$$\left( {x - y} \right)\widehat i + \left( {y - x} \right)\widehat j + \left( {x + y + z} \right)\widehat k$$is Stokes theorem connects The expression curl$$\left( {grad\,f} \right)$$where$$f$$is a scalar function is If$$\overrightarrow V $$is a differentiable vector function and$$f$$is sufficienty differentiable scalar function then curl$$\left( {f\overrighta...

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