1
GATE EE 2013
+2
-0.6
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
A
$$4xy{a_x} + 6yz{a_y} + 8zx{a_z}$$
B
$$4{a_x} + 6{a_y} + 8{a_z}$$
C
$$\left( {4xy + 4{z^2}} \right){a_x} + \left( {2{x^2} + 6yz} \right){a_y} + \left( {3{y^2} + 8zx} \right){a_z}$$
D
$$0$$
2
GATE EE 2013
+2
-0.6
Given a vector field $$\overrightarrow F = {y^2}x\widehat a{}_x - yz\widehat a{}_y - {x^2}\widehat a{}_z,$$ the line integral $$\int {F.dl}$$ evaluated along a segment on the $$x-$$axis from $$x=1$$ to $$x=2$$ is
A
$$2.33$$
B
$$0$$
C
$$-2.33$$
D
$$7$$
3
GATE EE 2012
+2
-0.6
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 $$K$$ is constant. The value of $$n$$ for which $$\nabla .A = 0\,\,$$ is
A
$$-2$$
B
$$2$$
C
$$1$$
D
$$0$$
4
GATE EE 2009
+2
-0.6
$$F\left( {x,y} \right) = \left( {{x^2} + xy} \right)\,\widehat a{}_x + \left( {{y^2} + xy} \right)\,\widehat a{}_y.\,\,$$ Its line integral over the straight line from $$(x, y)=(0,2)$$ to $$(x,y)=(2,0)$$ evaluates to
A
$$-8$$
B
$$4$$
C
$$8$$
D
$$0$$
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