GATE EE
Engineering Mathematics
Vector Calculus
Previous Years Questions

## Marks 1

Let $$f(x,y,z) = 4{x^2} + 7xy + 3x{z^2}$$. The direction in which the function f(x, y, z) increases most rapidly at point P = (1, 0, 2) is
Let $$\overrightarrow E (x,y,z) = 2{x^2}\widehat i + 5y\widehat j + 3z\widehat k$$. The value of $$\mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern-5.5... The value of line integral$$\,\,\int {\left( {2x{y^2}dx + 2{x^2}ydy + dz} \right)\,\,} $$along a path joining the origin$$(0, 0, 0)$$and the point... Let$$\,\,\nabla .\left( {fV} \right) = {x^2}y + {y^2}z + {z^2}x,\,\,$$where$$f$$and$$V$$are scalar and vector fields respectively. If$$V=yi+zj...
The line integral of function $$F=yzi,$$ in the counterclockwise direction, along the circle $${x^2} + {y^2} = 1$$ at $$z=1$$ is
The two vectors $$\left[ {\matrix{ {1,} & {1,} & {1} \cr } } \right]$$ and $$\left[ {\matrix{ {1,} & {a,} & {{a^2}} \cr ... Divergence of the$$3-$$dimensional radial vector field$$\overrightarrow r $$is Divergence of the vector field$$v\left( {x,y,z} \right) = - \left( {x\,\cos xy + y} \right)\widehat i + \left( {y\,\cos xy} \right)\widehat j + \lef...
Given a vector field $${\overrightarrow F ,}$$ the divergence theorem states that
The directional derivative of $$f\left( {x,y} \right) = 2{x^2} + 3{y^2} + {z^2}\,\,$$ at point $$P\left( {2,1,3} \right)\,\,$$ in the direction of the...

## Marks 2

The line integral of the vector field $$\,\,F = 5xz\widehat i + \left( {3{x^2} + 2y} \right)\widehat j + {x^2}z\widehat k\,\,$$ along a path from $$(... Match the following. List-$${\rm I}P.$$Stoke's Theorem$$Q.$$Gauss's Theorem$$R.$$Divergence Theorem$$S.$$Cauchy's Integral Theorem List-... The curl of the gradient of the scalar field defined by$$\,V = 2{x^2}y + 3{y^2}z + 4{z^2}x$$is Given a vector field$$\overrightarrow F = {y^2}x\widehat a{}_x - yz\widehat a{}_y - {x^2}\widehat a{}_z,$$the line integral$$\int {F.dl} $$evalu... The direction of vector$$A$$is radially outward from the origin, with$$\left| A \right| = K\,{r^n}$$where$${r^2} = {x^2} + {y^2} + {z^2}$$and ...$$F\left( {x,y} \right) = \left( {{x^2} + xy} \right)\,\widehat a{}_x + \left( {{y^2} + xy} \right)\,\widehat a{}_y.\,\,$$Its line integral over the... for the scalar field$$u = {{{x^2}} \over 2} + {{{y^2}} \over 3},\,\,$$the magnitude of the gradient at the point$$(1,3) is
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