GATE CE
Engineering Mathematics
Vector Calculus
Previous Years Questions

## Marks 1

The divergence of the vector field $$\,V = {x^2}i + 2{y^3}j + {z^4}k\,\,$$ at $$x=1, y=2, z=3$$ is ________.
For the parallelogram $$OPQR$$ shown in the sketch. $$\,\overrightarrow {OP} = a\widehat i + b\widehat j$$ and $$\,\overrightarrow {OR} = c\wideha... If$$\overrightarrow a $$and$$\overrightarrow b $$are two arbitrary vectors with magnitudes$$a$$and$$b$$respectively,$${\left| {\overrightarro...
For a scalar function $$f(x,y,z)=$$ $${x^2} + 3{y^2} + 2{z^2},\,\,$$ the gradient at the point $$P(1,2,-1)$$ is
The vector field $$\,F = x\widehat i - y\widehat j\,\,$$ (where $$\widehat i$$ and $$\widehat j$$ are unit vectors) is
For the function $$\phi = a{x^2}y - {y^3}$$ to represent the velocity potential of an ideal fluid, $${\nabla ^2}\,\,\phi$$ should be equal to zero....
The directional derivative of the function $$f(x, y, z) = x + y$$ at the point $$P(1,1,0)$$ along the direction $$\overrightarrow i + \overrightarro... The derivative of$$f(x, y)$$at point$$(1, 2)$$in the direction of vector$$\overrightarrow i + \overrightarrow j $$is$$2\sqrt 2 $$and in the ... ## Marks 2 The directional derivative of the field$$u(x, y, z)={x^2} - 3yz$$in the direction of the vector$$\left( {\widehat i + \widehat j - 2\widehat ...
A particle moves along a curve whose parametric equations are: $$\,x = {t^3} + 2t,\,y = - 3{e^{ - 2t}}\,\,$$ and $$z=2$$ $$sin$$ $$(5t),$$ where $$x... For a scalar function$$\,f\left( {x,y,z} \right) = {x^2} + 3{y^2} + 2{z^2},\,\,$$the directional derivative at the point$$P(1,2,-1)$$in the direct... The velocity vector is given as$${\mkern 1mu} \vec V = 5xy\widehat i + 2{y^2}\widehat j + 3y{z^2}\widehat k.{\mkern 1mu} {\mkern 1mu} $$The divergen... The directional derivative of$$\,\,f\left( {x,y,z} \right) = 2{x^2} + 3{y^2} + {z^2}\,\,$$at the point$$P(2,1,3)$$in the direction of the vector... Value of the integral$$\,\,\oint {xydy - {y^2}dx,\,\,} $$where,$$c$$is the square cut from the first quadrant by the line$$x=1$$and$$y=1$$will... The line integral$$\int {\,\,V.dr\,\,} $$of the vector function$$V\left( r \right) = 2xyz\widehat i + {x^2}z\widehat j + {x^2}y\widehat k\,\,$$fro... The directional derivative of the following function at$$(1, 2)$$in the direction of$$(4i+3j)$$is :$$f\left( {x,y} \right) = {x^2} + {y^2}
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