1
GATE EE 2007
+1
-0.3
Divergence of the vector field $$v\left( {x,y,z} \right) = - \left( {x\,\cos xy + y} \right)\widehat i + \left( {y\,\cos xy} \right)\widehat j + \left[ {\left( {\sin {z^2}} \right) + {x^2} + {y^2}} \right]\widehat k\,\,$$
A
$$2z\,\cos {z^2}$$
B
$$\,\sin \,xy + 2z\,\cos {z^2}$$
C
$$x\,\sin xy - \cos z$$
D
none of these
2
GATE EE 2002
+1
-0.3
Given a vector field $${\overrightarrow F ,}$$ the divergence theorem states that
A
$$\int\limits_s {\overrightarrow F .d\overrightarrow s = \int\limits_v \nabla .\overrightarrow F \,dv}$$
B
$$\int\limits_s {\overrightarrow F .d\overrightarrow s = \int\limits_v \nabla \times \overrightarrow F \,dv}$$
C
$$\int\limits_s {\overrightarrow F \times d\overrightarrow s = \int\limits_v \nabla .\overrightarrow F \,dv}$$
D
$$\int\limits_s {\overrightarrow F \times d\overrightarrow s = \int\limits_v \nabla \times \overrightarrow F \,dv}$$
3
GATE EE 1994
+1
-0.3
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 vector $$\,\,a = \overrightarrow i - 2\overrightarrow k \,\,$$ is
A
$$4/\sqrt 5$$
B
$$-$$ $$4/\sqrt 5$$
C
$$\sqrt 5 /4$$
D
$$-$$ $$\sqrt 5 /4$$
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